CHAPTER 2 SCALAR AND VECTOR

CONTAIN:1. Scalar & vector.2. Multiplication of a vector by number 3. Addition of vectors by graphical method & properties4. Kinds of vector5. Resolution of a vector 6. Composition of a vector. 7. Addition of vectors by rectangular components method. 8. Product of two vectors 9. Properties of dot product & Special cases of dot product10. Equations 11. Dimensions 12. Shot questions and answers

TECHNECHAL RELATIVE TERM DEFINITIONS AND EQUATIONS:

Physical quantities: A physical quantity is expressed as the product of a numerical value

and a physical unit, not only by a number. It does not depend on the unit distance (1 km is the same as 1000 m),

although the number depends on the unit. Most of the physical quantities used in physics are either scalar or vector quantities.

Scalar: Scalar quantities are numbers that have a magnitude but no direction.

The examples of scalar quantities are time, speed, temperature, and volume. A scalar quantity no directional component, has only magnitude.  For example, the units for time (minutes, days, hours, etc.) represent an amount of time only and tell nothing of direction. All arithmetical rules are applicable on scalars.

Vector: Vectors are a geometric way of representing quantities that have direction as well as magnitude.

A vector notation that should be used is with an arrow over a letter, such as . The magnitude of a

vector is denoted by absolute value signs around the vector symbol: magnitude of the vector is

written as either A = . Graphically, a vector is represented by an arrow, defining the direction, and the length defines the vector's magnitude. Such representative line connecting an initial point A with a

terminal point B, and denoted by .

Multiplication of a vector by number (0r scalar):Statement:A vector multiplied with any positive or negative number (or scalar), it gives a product vector. A process to determine such product vector is called “Multiplication of a vector by a number (or scalar)”.

Explanation:

Suppose vector is to be multiplied by “+ n” number. The product vector is + n . The magnitude

product quantity is n having same direction.

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And a vector multiplied by “-n”, the product vector is - n . The magnitude of new generated product

vector quantity has n in the opposite direction. Commutative law for multiplication:

By changing the order of vector no effect on the product vector, such phenomenon is called “Commutative property”.

m = m Associative law for multiplication:

By changing the pair of product between two numbers (or vector with number) with other respective number or vector, shows no effect on product the product quantity is called “Associative property”.

m n ( ) = n (m ) = (m n ). Distributive law for multiplication:

Distribution of a vector among numbers no effect on the product vector or a number among vector. Such phenomenon is called “Distributive property”.

(m + n) = m + n

OR . Division of vector by number:

The division of a vector by a number is the process of multiplication of a vector by a number to generate new vector quantity of different magnitude. Such type of process called “Division of a vector by number”.

, Because,

Addition of vectors by graphical method:Statement:A process to join number of vectors to determine a resultant vector is called “Addition of vectors”. Such single new vector (or resultant vector) has combined effect (or sum) of all the vectors.

Explanation: The resultant vector cannot be determined by ordinary mathematical rule. Such single new vector is determined graphically by Initial to Terminal rule. According to this rule, “The vectors are joined in such a way that Initial of Second vector coincide the Terminal of First vector, the Initial of Third vector coincide the Terminal of Second vector, and join other vectors in the same way. In the last join the Initial of First vector directed Terminal of Last Vector”.

Commutative property for addition of vectors:Statement: By changing the order of vectors, there is no effect on the magnitude and direction of the resultant vector. It is known as “Commutative law” for addition of vectors.

Explanation:

Suppose two vectors and are the adjacent sides of a parallelogram in terms of magnitude and direction. Complete the parallelogram ABCD. Join the initials of

two vectors directed their terminals. We get resultant vector as diagonal in

terms of magnitude and direction, representative line .

Where, (in ABC triangle)

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And (in ADC triangle)

And (in terms of magnitude and direction) It is the common diagonal; as resultant of both triangles with same magnitude and direction.

Therefore, , Commutative property is valid for addition of vectors

Associative property for addition of vectors: Statement:By changing the given vectors sum (Resultant), there is no effect on the magnitude and direction of resultant vector. It is known as “Associative law” for addition of vectors.

Explanation:

Suppose three vectors , and are joined by initial to terminal rule. Hence, is the resultant, has combined effect of three added vectors.

Join and , we get .

Therefore,

Now join and ,we get

Hence,

It means, , in terms of magnitude and direction

Therefore , Associative property is valid for addition of vectors.

Addition of vectors by law of parallelogram method:Two vectors acting at a point are represented in terms of magnitude and direction of sides of parallelogram. The resultant is represented by the diagonal of the parallelogram in terms of magnitude and direction.

Suppose two vectors and are acting at a common point “O”, have

representative lines and respectively, as two sides of parallelogram in terms of magnitude and direction. Complete the parallelogram OACB, by shifting the

vectors. Draw a diagonal represents in terms of magnitude and direction the resultant vector .

Cosine law for magnitude.

Sine law for direction.

Subtraction of two vectors:

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Vector subtraction is defined in the following way. The difference of two vectors, - , is a vector

that is, = Thus vector subtraction can be represented as a vector addition. The graphical represented, that place

the initial point of the vector - on the terminal point the vector , and then draw a line from the

initial point of to the final point of - to give the difference .

Kinds of vectors:1. Parallel vector or Equal vector: Vectors are equal if they have the same magnitude and direction.

Two vectors have equal magnitude and same direction regardless of their

initial points. Thus one vector is equal to other vector, . These two are equal vectors. The angle made between two same directed vectors (or parallel vectors) is 0o.

2. Negative vector or Opposite vector: The negative of a vector has the same magnitude but opposite direction Two vectors have equal

magnitude and opposite direction regardless of their initial points.

Thus, one vector is negative to other vector, . The angle made between two opposite directed vectors is 180o.

3. Free vector: A vector sifted from one point to another with out changing magnitude and direction

4. Null vector: A vector of zero magnitude is called “null vector” or zero vectors. A vector at origin has no any

direction is the null vector. 5. Unit vector: A unit vector is a vector pointing in a given direction with a magnitude of

one. A unit vector is represented by a cap (^) symbol above the of a vector.

Suppose a vector having rectangular coordinates , and

along x-, y- and z- axes respectively. The vectors, with respect to the reference axes can be written as,

The magnitude of a vector is,

6. Position vector:

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A vector at fixed reference position point having magnitude and direction is called “Position

vector”. Thus, every vector is the position vector. Suppose “P” is a position with respect to an

origin point “O”. So that is the position vector, denoted by “ ” having rectangular

coordinates, Ax, AY and Az given by unit vectors , and respectively.

Therefore, , is the position vector.

, is the magnitude.

DESCRIPTIVE PART1. Determination of rectangular components (or Resolution of a vector): Statement:A vector resolved in to two vectors making an angle of 90o with each other such process is known as determination of Rectangular Components of a vector or called resolution of a vector (or decomposition of a vector).

Explanation:

Suppose a vector is lying in the x-y plane and making an angle with the

positive x axis, has representative line . Draw perpendiculars on to X and Y

axes, from the terminal point of a given vector. We get component

represents the projection of along the x axis, and the component

represents the projection of along the y axis with representative lines

and respectively. They are called rectangular components of a vector. Add

and according initial to terminal rule, the angle between them is 90o.

Mathematical Derivation:The magnitudes of rectangular components are to be determined by, Consider OQT right angled triangle,

2. Determination of a vector (or composition of a vector):Definition:A collective effect of rectangular components of a vector as the resultant vector such process is known as determination of a vector or composition of vector.

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Explanation:

Consider two vectors and making an angle of 90o with +X axis with representative lines and

respectively. Add according initial to terminal rule, gives the resultant vector . The length of the gives magnitude and the tip of the arrow indicate the direction of a vector.

Mathematical Derivation:The magnitude and direction of resultant vector is to be determined by, Consider OPS right angled triangle,

(OS)2 = ( OP ) 2 + ( PS ) 2

and

3. Addition of vectors by rectangular components method:Addition of vectors is a process, to determine a Resultant vector. Such a resultant vector has the same effect as sum of all the vectors added together.

Explanation and Mathematical derivation:

Suppose and are two vectors making angles 1 and 2 with respect to x-axis. Add such vectors according initial to terminal rule.

We get a resultant vector . Resolve the given vectors in to rectangular components. We get along x- axis,

It means, the magnitude of X - component of the resultant vector (Rx), is equal to the sum of magnitudes of X- components of given vector. And, we get along y- axis,

It means, the magnitude of Y-component of the resultant vector (Ry), is equal to the sum of magnitudes of Y-components of given vector.After finding the magnitudes of X-and Y-components of a resultant vector,

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We can find the magnitude of resultant vector by formula,

And in the last find the direction of resultant vector 4. Product of two vectors:There are two different kinds of vector products that are used:

1. Scalar Product: yields a scalar as the product; commonly called the dot product. 2. Vector Product: yields a vector with a direction perpendicular to the plane formed by the two

vectors being multiplied; commonly called the cross product

i. Scalar Product or dot product: Statement:Two vector quantities multiplied together and their product is scalar quantity, such type of product is called “Scalar Product”. In this type of product place a dot (.) between two vectors, so that scalar product is also called “Dot product”.

Explanation and Mathematical derivation:

Suppose two vectors, and , separated by an angle of between them.

The scalar product of two vectors is written as . Its magnitude written

as,

Now, in the illustration to the right, Vector can be split into its components. The component B cos is

parallel to . Commutative, distributive and associative properties are valid.

Example:

Work done is Scalar product. If force produces a displacement then,

Work =

= F d Cos Similarly, Power is a scalar quantity is another example of dot product. It is dot product between force

and velocity .

Hence Power = = F v Cos

ii. Cross Product or Vector Product:Statement:Two vector quantities are multiplied and their product is Vector quantity, such type of product is called “Vector Product”. In this type of product place a cross ( ) between two vectors, so that vector product is also called “cross product”.

Explanation and Mathematical derivation:

Suppose two vectors, and , separated by an angle of between them. The vector product of two

vectors is written as . Its magnitude written as,

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Commutative is not valid but distributive and associative properties are valid.

Example:

Example, torque is the vector product of force and the moment arm then,

=

= = F r Sin, as magnitude of torque.Similarly, Angular momentum is a vector quantity is another example of vector product. It is cross

product between linear momentum and moment arm .

Hence (Angular momentum) = = = p r Sin ,

The cross product gives a vector quantity with a direction perpendicular to the plane of the two vectors and a magnitude given by the above equation.

The Vector product of two vectors is written as = A B Sin .

The direction the vector is determined by a Right hand rule. This rule states that,

“Imagine rolling vector around the perpendicular vector towards . Now curl the

fingers of right hand so they trace the imaginary path of vector . Stick thumb out and the direction of the vector is the same direction your thumb is facing. Anti-clock wise rotation of fingers indicates the direction of new vector along positive z- axis and clock wise along negative z-axis. (clockwise is inward; counterclockwise is outward) ”.

5. Algebraic Properties of dot product. i. Commutative law for Dot product:

Statement:The scalar product of two vectors does not change with the change in the order of vectors to be multiplied.

Explanation and Mathematical Derivation:

Suppose, and are two vectors in their respective direction. Resolve the vector in the direction of . Resolve

the vector , into its rectangular components. We get, . = (A Cos ) B

And resolve the vector in to its rectangular components. We get,

. = (B Cos ) A Hence, (A Cos ) B = ( B Cos ) A

Therefore, . = . Thus, the scalar product of two vectors does not change with the change in the order of vectors to be multiplied.This shows that commutative law for dot product is applicable.

ii. Distributive law for dot product: Statement:To distribute vector among two or more vectors no effect on the dot product is called distributive property.

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Explanation and Mathematical Derivation:

Let us consider three vectors , and . Draw the projections from vectors

and on to the direction of vector .

Projection from onto direction of = Projection from onto

direction of + Projection from onto direction of .

By multiplying both sides by , we get,

This shows that distributive law for dot product is valid.

iii. Associative law for dot product:

This shows that, Associative law for dot product is valid.

iv. Special cases of dot product:

i. Show that , Proof:

We know that

And = 0o

Hence, shown.

Similarly,

ii. Show that , Proof:

We know that

And = 0o

Hence, shown

iii. Show that ,

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Proof:

We know that

And = 180o

Hence, shown

iv. Show that , Proof:

We know that

Hence, shown

v. Show that , Proof:

We know that

Hence, shown

Similarly,

And,

We get,

vi. Show that , Proof:

We know that

Hence, shown

Similarly,

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And,

We get,

vii. Show that , R =Proof:

We know that Suppose,

Hence R = shown

Similarly, R =

6.Algebraic Properties of Cross Product

i. Commutative property:

To find put the fingers of right hand on first vector curl towards second vector .The new

vector is along positive Z-axis.

=

And to find put the fingers of right hand on first vector curl

towards second vector. The new vector is along the negative Z- axis.

Thus,

Hence,

The cross product is anti commutative

Therefore,

This shows that, Commutative property is not valid for cross product, because by changing the order of vectors the direction of cross product becomes reverse.

ii. Distributive property:

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This shows that, Distributive property for cross product is valid.iii. Associative property:

This shows that, Associative property for cross product is valid.

Special cases of cross product:i. Show that ,

(Say, null vector)

Where, And Proof:Two vectors are in the same direction, say = 0o

Therefore,

Or Hence shown

ii.

iii.

iv.

v.

vi.

vii.

viii.

ix. If , and

Therefore,

Then,

x. Area of parallelogram = = A h = AB sin

xi. Area of triangle = (Area of parallelogram)= = A h= (AB sin )

Equations

1. 2.

3. 4.

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5. 6.

7. 8.

9. = A B Cos 10. = A B Sin

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26. . = .

27.

SHORT QUESTIONQ: No.1. Is heat vector quantity?Heat is merely the measure of how much an atom is vibrating. Vibrations have no direction and as such heat cannot be a vector.

Q: No.2. Which type of vectors ate added or subtracted.Vectors must have the same units in order for them to be added or subtracted

Q: No.3. Electric current is scalar or vector quantity?It’s a scalar quantity. But in circuits we define that direction of current is direction opposite to the motion of electrons. It doesn’t means that it is a vector; it represents only direction of rate of flow of charge.

Q: No.4. Is Area a scalar or vector quantity?Usually area is a scalar quantity. In more advanced calculus area is vector. Area is a vector because as

we know which is scalar,

,

Which is, = vector, so area is a vector.

In geometry, for a finite planar surface of scalar area S, the vector area is defined as a vector whose magnitude are S and whose direction is perpendicular to the plane, as determined by the right hand rule.

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Q: No.5. What you mean by negative vector?A vector whose magnitude is same as that of given vector but direction is opposite.

Q: No.6. What is the importance of scale in representation a vector?We have to represent a vector on a limited space of a paper. Vectors with large magnitude are not possible to draw on a page. So scale is used to represent a vector.

Q: No.7. Why we define a null vector though it has no magnitude and it may be in any arbitrary direction?Because, a null vector has a zero magnitude and has any arbitrary direction but it has too much importance. As we know any vector has magnitude and direction. Under certain condition a vector has magnitude and direction is shown by a unit vector. Hence this vector is always used to indicate the direction of vector.

Q: No.8. If is the resultant vector of vectors and , can we say and are the components of

vector .

If the angle made between and , is of 90o then the vectors are the rectangular components of a

resultant vector . If the angle between and is of not 90o, then the vectors are not rectangular

components of a vector . But vector is resultant vector, having combined effect of two vectors.

Q: No.9. Can the magnitude of a vector is negative?No, the magnitude of a vector never is negative. The direction of a vector is negative.

Q: No.10. Can the components of a vector are negative? Yes, a component of a vector is negative.

Q: No.11. How is the addition of vectors different from the addition of scalars?The addition of vectors is different from addition of scalars. Because, vectors have magnitude and direction. So that vectors are added by initial to terminal rule or by resolution of vectors or by parallelogram method. These methods are used to find out a single new vector called resultant vector has collective effect of all the vectors. On the other side scalar have only magnitude, so that they are added according to mathematical rule.

Q: No.12. What is the difference between the components of a vector and the magnitude of a vector? Is it possible for the magnitude to equal the value of a component? When could this occur? The components of a vector and magnitude of a vector are different from one another. The components of a vector are always making an angle of 90o with each other. The magnitude of a vector is length of line of a vector. Yes, it is possible that the magnitude equal to the value of component. This is possible only when the vector is making angle of zero degree or 90o then the magnitude equal to the value of component.

Q: No.13. Distinguish between scalar and vector.Vector quantities have two characteristics, a magnitude and a direction. Scalar quantities have only a magnitude. When comparing two vector quantities of the same type, we have to compare both the magnitude and the direction. For scalars, we only have to compare the magnitude.

Q: No.14. Can force directed north balance a force directed east? No, force directed north couldn’t balance a force directed east. Because, these forces are at right angle, each other.

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Q: No.15. Time is directed from present to future. Is it a vector quantity?You said it right there. Time can only go forward. Think about it this way. In English, the only way we could quantify time is to refer to it as a "length" of time. Length is a scalar quantity. There is no directionality associated with time's magnitude; it is only duration, and that is why we refer to it as a scalar. Time is a scalar quantity, because present to future does not represent any definite specified direction.Q: No.16. Can we talk a vector quantity of zero magnitude?Yes, we can talk a vector of zero magnitude. In vector algebra we say in tem of “Null vector”.

Q: No.17. Is it possible to combine two vectors of different magnitudes to give zero resultant? No, it is not possible to combine two vectors of different magnitudes to give zero resultant.

Q: No.18. Is it possible to add three vectors of equal magnitudes but different directions to get zero resultant?Yes, it is possible to add three vectors of same magnitudes but in different directions to get zero resultant. They are arranged in such a way that three vectors of same magnitudes to make an equilateral triangle. The resultant (one vector of three as one side of a triangle) will be null vector of zero magnitude.

Q: No.19.Can the magnitude of resultant of two vectors be greater than the sum of magnitude of the individual vectors?No, the resultants of two vectors have magnitude either equal to or less than the sum of magnitude of individual vectors.

Q: No.20. Can a scalar product of two vectors be negative?Yes, it is possible that dot product of two vectors be negative. The vectors are opposite to one another, making an angle of 180o with each other.

. = A B Cos 180o

. = - A B Work can be negative when force and displacement are in opposite direction.

Q: No.21. If . = 0, can it be conclude that and are perpendicular to each other.

If dot product between or is equal to zero magnitude vector.

It can be concluding that and are perpendicular to one another. Say . = A B Cos 90o

Q: No.22. If = 0, can it be conclude that and are parallel to one another (or anti parallel) to each other.

If cross product between and is equal to zero magnitude vector, say null vector.

It can be conclude that and are parallel, say, =.0o and when vectors anti parallel then say, = 180o

Or

Q: No.23. What do you mean by negative vector?Two vectors of same magnitudes but opposite directions are known as negative vectors.

Q: No.24. Explain multiplication of a vector by a number. Any vector can be multiplied with positive or negative number. The product of them will be a new vector. Such product vector has same direction then multiplication is with positive number and the direction is opposite then multiplication is with negative number.

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Q: No.25. What do you mean by rectangular components?When the two vectors are at right angle (say 90o), then the vectors known as “rectangular components”.

Q: No.26. What do you men resolution of a vector?A process of splitting a vector, in rectangular components is called “resolution of a vector”.

Q: No.27. What do you mean by resultant vector?A single vector have combined effect, of number of vectors which are to be added, is called “resultant vector”. It is sum of number of vectors.

Q: No.28. Define scalar and vector. The non-directional quantities are scalars. It means Scalars have only a magnitude. And vectors are directional quantities. It means vectors have magnitude and specified direction.

Q: No.29. How can a vector be resolved in to rectangular components? A vector can be resolved in to rectangular components by drawing perpendiculars from the terminal of a given vector, on to X and Y-axes. We get one component along X-axis while the other along Y-axis. The angle made between them is 90o.

Q: No.30. Define unit vector.

A vector of magnitude only one, is called “unit vector”. Q: No.31. Define position vector A vector at any position point with respect reference origin, is called “position vector”

Q: No.32. Define null vector.A vector of zero magnitude is called “null vector’.

Q: No.33. Which of the following is a vector quantity i) Density ii) energy iii) temperature iv) Centrifugal force v) Displacement vi) distance vii) length viii) position

The vector quantities are, i) Centrifugal force ii) Displacement iii) Position

Q: No.34. What is the condition for the two vectors to be, i. Parallel ii. Perpendicular.i. When the angle between two vectors is zero degree then they are parallel to one an anotherii. When the angle between two vectors is ninety degree then they are perpendicular to one another.

Q: No.35. . = . , then = + , is it true or false?

Yes, it is true. When, = + , then two vectors are parallel and angle between them is zero degree.

Hence, = A B Cos

= A A Cos0o = A2

Similarly, = B B Cos 0o = B2

A2 = B2

Therefore,

Q: No.36. What do you conclude, if the component of along is zero?

If the component of along is zero; then and are perpendicular on each other.

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Q: No.37. And , find the angle between the vectors, if and are none zero vectors. Answer:

We know that

Given

Therefore,

Hence, proved

And, we know that,

Hence proved

Q: No.38. Dot product of two vectors is a scalar quantity; you give at least two examples.We know that, work is the dot product between force and displacement. It is scalar quantity.

Work = And power is also scalar quantity it is dot product between force and velocity.

Power =

Q: No.39. Find the direction of a vector, if x- component is 10 and y-component is 16.Answer:

= Tan-1 (1.6) = 58o

Q: No.40. Show that, i) and ii)

Answer:

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i) We know that

, Two unit vectors are parallel to one another; hence angle made between them is 0o.

Thus,

, hence shown.

i) we know that

, Two unit vectors are perpendicular to one another; hence angle made between them is 90o.

Q: No.41. Show that, consider only magnitude of .Answer:

We know that,

And

, as magnitude

, hence provedL H S = R H S

Q: No.42. If the dot product between two vectors is zero, then what is the angle between them?When dot product between two vectors is zero then the angle made between two vectors is 90o.

Q: No.43. If is the vector sum of and , does have to lie in the plane determined by and .

= + , this shows that lies in the plane of and .

Q: No.44.Supposing is none zero vector .If . = 0 and also = 0, where is an unknown vector.

What can you conclude that about ? Answer:

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. = 0

And = 0

Given is non zero, then it is conclude that is zero.

Q: No.45. Can the magnitude of resultant vector be greater than the sum of magnitude of individual vector?No, it is not possible that the magnitude of resultant vector can be greater than the sum of magnitude of individual vector.

Q: No.46. Vector lies in the xy –plane, for what orientation will both of its rectangular components be negative? For what orientation will its components have opposite signs?

i) If a vector lies in 3rd quadrant, it makes an angle 180o 270o, then both rectangular components will be negative.

ii) If a vector lies in 2nd quadrant, it makes an angle 90o 180o, then horizontal component will be negative and vertical positive.

iii) If a vector lies in 4th quadrant, it makes an angle 270o 360o, then horizontal component will be positive and vertical negative.

Q: No.47. If one of rectangular components of a vector is not zero, can its magnitude be zero? No, its magnitude never is zero.

Q: No.48. Under what condition a vector have components that are equal in magnitude?In case of rectangular components, the magnitude of the two components will be equal if the vector makes an angle of 45o with +x axis

Q: No.49. Can you add zero to a null vector?No, zero cannot be added to null vector, as one i.e. zero is scalar and other is a vector.

Q: No.50. What are the properties of dot product. Write few properties. Properties of the Dot Product

1)  2) 3)

4)

Q: No.51. What are the properties of cross product? Write few properties.

thenThe cross product of any two parallel vectors is the null vector since sin 0 = 0, and also 

 Using these, we can eventually find: 

That's our equivalent definition. If you're familiar with determinants you may see this can be written more conveniently as,

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= Some Properties of the Cross Product

The cross product is anti commutative: 

The cross product is distributive: The cross product of parallel vectors is the null vector, in particular: 

Also is the area of the parallelogram formed by   and ,

Q: No.52, What is a vector? A vector is a physical measurement which has size, units and direction. Many properties of matter are dependent on direction, for example displacement, and the direction from the point of measure is given. These properties are known as vectors. Some properties of matter do not depend on direction; these have only size and units (and not always units). The name of this type of property is a scalar. An example of a scalar is time, whether in the north, south, east or west time passes the same and is independent of direction. Note that the numeric and unit portion of a vector are called the magnitude, and its direction is expressed as a compass angle.

Q: No.53. Two equal forces act at a point. The angle between their lines of action is zero degree. Find the magnitude and direction of the resultant force.Two equal forces act at a point. The angle between their lines of action is zero degree. The magnitude become double and direction will be same of the resultant force

Q: No.54. What is the difference between cross product and dot product why can't we calculate work using cross product?The vector cross product gives another vector which is normal to both of the original vectors, with a magnitude equal to the area subtended by the two. The dot product gives a scalar. Work, being energy, is a scalar, not a vector. That's why you cannot calculate work using a cross product.

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