vectors

VectorsAdvanced Level Pure Mathematics

Advanced Level Pure Mathematics

Algebra

Chapter 7 Vectors

7.1 Fundamental Concepts 27.2 Addition and Subtraction of Vectors 27.3 Scalar Multiplication 37.4 Vectors in Three Dimensions 57.5 Linear Combination and Linear Independence 77.6 Products of Two Vectors 13

A. Scalar ProductB. Vector Product

7.7 Scalar Triple Product 22Matrix Transformation* 24

7.1 Fundamental Concepts

1. Scalar quantities: mass, density, area, time, potential, temperature, speed, work, etc.

2. Vectors are physical quantities which have the property of directions and magnitude.Prepared by K. F. Ngai

7


e.g. Velocity , weight , force , etc.

3. Properties:

(a) The magnitude of is denoted by .

(b) if and only if , and and has the same direction.

(c)

(d) Null vector, zero vector , is a vector with zero magnitude i.e. .

The direction of a zero vector is indetermine.

(e) Unit vector, or , is a vector with magnitude of 1 unit. I.e. .

(f)

7.2 Addition and Subtraction of Vectors

1. Geometric meaning of addition and subtraction.

2. Properties: For any vectors and , we have

(a) ,(b) ,

(c)

(d)

N.B. (1)

(2)

7.3 Scalar Multiplication

When a vector is multiplied by a scalar m, the product is a vector parallel to a such that(a) The magnitude of is times that of .

(b) When , has the same direction as that of ,Prepared by K. F. Ngai

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When , has the opposite direction as that of .

These properties are illustrated in Figure.

Theorem Properties of Scalar Multiplication

Let be two scalars. For any two vectors and , we have(a)

(b)

(c)

(d)

(e)

(f)

Theorem Section Formula

Let A,B and R be three collinear points.

If , then .

Example Prove that the diagonals of a parallelogram bisect each other.

Solution

Prepared by K. F. Ngai


Properties

(a) If are two non-zero vectors, then if and only if for some .

(b) , and

7.4 Vectors in Three Dimensions

(a) We define are vectors joining the origin to the points , , respectively.

(b) and are unit vectors. i.e. .

(c) To each point in , there corresponds uniquely a vector

where is called the position vector of .

(d)

(e)

(f) Properties : Let and . Then

(i) if and only if and ,

(ii)

(iii)

N.B. For convenience, we write

Example Given two points and .Prepared by K. F. Ngai


(a) Find the position vectors of and . .

(b) Find the unit vector in the direction of the position vector of .

(c) If a point P divides the line segment in the ration , find the coordinates of .

Solution

Example Let and

(a) Find the position vectors of and . Hence find the length of .

(b) If is a point on such that find the coordinates of .

(c) Find the unit vector along .

Solution



7.5 Linear Combination and Linear Independence

Definition Consider a given set of vectors A sum of the form

where are scalars, is called a linear combination of

If a vector can be expressed as

Then is a linear combination of .

Example is a linear combination of the vectors .

Example Consider , show that is a linear combination of and

while is not.

Solution



Definition If are vectors in and if every vector in can be expressed as the linear

combination of . Then we say that these vectors span (generate) or

is the set of the basis vector.

Example is the set of basis vectors in .

Example is the set basis vector in .

Remark : The basis vectors have an important property of linear independent which is defined as follow:

Definition The set of vector is said to be linear independent if and only if the vectors

equation has only solution

Definition The set of vector is said to be linear dependent if and only if the vectors

equation has non-trivial solution.

(i.e. there exists some such that )

Example Determine whether are linear independent or

dependent.

Solution



Example Let and Prove that

(a) and are linearly independent.

(b) any vector in can be expressed as a linear combination of and .

Solution



Example If vectors and are linearly independent, show that and are also linearly

independent.

Solution

Example Let , and

(a) Show that and are linearly independent for all real values of .

(b) Show that there is only one real number so that , and are linearly dependent.

For this value of , express as a linear combination of and .

Solution



Theorem

(1) A set of vectors including the zero vector must be linearly dependent.

(2) If the vector can be expressed as a linear combination of , then the set of vectors

and are linearly dependent.

(3) If the vectors are linearly dependent, then one of the vectors can expressed as a linear

combination of the other vectors.

Proof



Example Let and .

Prove that and are linearly dependent.

Solution

Theorem Two non-zero vectors are linearly dependent if and only if they are parallel.

Proof



Theorem Three non-zero vectors are linearly dependent if and only if they are coplanar.

Proof

7.6 Products of Two VectorsA. Scalar Product

Definition The scalar product or dot product or inner product of two vectors and , denoted by , is

defined as

where is the angle between and .

Remarks By definition of dot product, we can find by .

Example If and angle between and is , then

Theorem Properties of Scalar Product

Let be three vectors and be a scalar. Then we have

(1)

(2)

(3)

(4)

(5) if and if

Theorem If and . Then



(1)

(2) =

=

(3) if and only if .

(4) if and only if .

Example Find the angle between the two vectors and

Solution

Remarks Two non-zero vectors are said to be orthogonal if their scalar product is zero. Obviously, two

perpendicular vectors must be orthogonal since , , and so their scalar product is

zero. For example, as and are mutually perpendicular, we have

.

Also, as and are unit vectors, .

Example State whether the two vectors and are orthogonal.

Solution

Example Given two points and

and two vectors

and

If is perpendicular to both and , find the values of and .

Solution



Example Let and be three coplanar vectors. If and are orthogonal, show that

Solution

Example Determine whether the following sets of vectors are orthogonal or not.

(a) and

(b) and

(c) and

Solution

B. Vector Product

Definition If and are vectors in , then the vector product and cross

product is the vector defined by

=

=

Example Find , and if and .

Solution



Example Let and Find

(a) (b) (c) (d)

(e) (f)

(g) (h)

(i) (j)

(k) (l)

Solution



Theorem If and are vectors, then

(a)

(b)

(c)

Proof



Remarks (i) By (c) =

= , where is angle between and .

=

=

=

The another definition of is where is a unit vector

perpendicular to the plane containing and .

(ii) and

(iii)

Definition The vector product (cross product) of two vectors and , denoted by , is a vector

such that (1) its magnitude is equal to , where is angle between a and b.

(2) perpendicular to both and and form a right-hand system.

If a unit vector in the direction of is denoted by , then we have

Geometrical Interpretation of Vector Product

(1) is a vector perpendicular to the plane containing and .

(2) The magnitude of the vector product of and is equal to the area of parallelogram with and as its

adjacent sides.

Corollary (a) Two non-zero vectors are parallel if and only if their vector product is zero.

(b) Two non-zero vectors are linearly dependent if and only if their vector product is zero.

Theorem Properties of Vector Product

(1)

(2)Prepared by K. F. Ngai


Example Find a vector perpendicular to the plane containing the points and

.

Solution

Example If show that

Solution

Example Find the area of the triangle formed by taking and as

vertices.

Solution

Example Let and .

(a) Find .

(b) Find the area of

Hence, or otherwise, find the distance from to .

Solution



Example Let and be two vectors in such that

and

Let .

(a) Show that for all

(b) For any , let Show that for all .

Solution

Example Let .

If , prove that .

Solution



Example Let and be linearly independent vectors in .

Show that :

(a) If , and ,

then

(b) If such that , then .

(c) If , then .

(d) If ,

then for all .

Solution

7.7 Scalar Triple Product

Definition The scalar triple product of 3 vectors and is defined to be .Prepared by K. F. Ngai


Let the angle between and be and that between and be .

As shown in Figure, when , we have

Volume of Parallelepiped =

=

=

=

=

=

Geometrical Interpretation of Scalar Triple Product

The absolute value of the scalar triple product is equal to the volume of the parallelepiped with

and as its adjacent sides.

Theorem Properties of Vector Product

Let , and be three vectors. Then

Remarks Volume of Parallelepiped =

Example Let ,

(a) Find the volume of parallelepiped with sides and .

(b) What is the geometrical relationship about point in (a).

Solution



Example are the points , respectively and is the origin.

Let and .

(a) Show that and are linearly independent.

(b) Find

(i) the area of , and

(ii) the volume of tetrahedron .

Solution

Matrix Transformation*

Linear transformation of a plane (reflections, rotation)

Consider the case with the point such that

=

= where



A is a matrix of transformation of reflection.

In general, any column vector pre-multiplied by a matrix, it is transformed or mapped into

another column vector.

Example ,

We have

If using the base vector in , i.e .

,

then can be found.

The images of the points under a certain transformation are known. Therefore, the

matrix is known.

Eight Simple Transformation

I. Reflection in x-axis

II. Reflection in y-axis

III. Reflection in .

IV. Reflection in the line



V. Quarter turn about the origin

VI. Half turn about the origin

VII. Three quarter turn about the origin

VIII. Identity Transformation

Some Special Linear Transformations on R 2

I. Enlargement

If then .

II. (a) Shearing Parallel to the x-axis

The y-coordinate of a point is unchanged but the x-coordinate is changed by adding to it a Prepared by K. F. Ngai


quantity which is equal to a multiple of the value of its y-coordinate.

(b) Shearing Parallel to the y-axis

III. Rotation

IV. Reflection about the line

Example If the point is rotated clockwise about the origin through an angle , find its final

position



Solution

Example A translation on which transforms every point whose position vector is

To another point with position vector defined by

Find the image of (a) the point

(b) the line

Solution

Linear Transformation

Definition Let and be two sets. A mapping is called a linear transformation from

to if and only if it satisfies the condition:

and

Example Let be the set of matrices and be any real matrix. A mapping

Such that . Show that is linear.

Solution



In , consider a linear transformation , let , .

We are going to find the image of under .

Therefore, can be found if and are known. That is to say, to specify completely, it is

only necessary to define and .

For instance, we define a linear transformation

by .

=

=

=

We form a matrix such that =

=

Consider = =

The result obtained is just the same as .

The matrix representing the linear transformation is called the matrix representation of the linear

transformation

Example Let , defined by

The matrix represent representation of a linear transformation is .

Example The matrix represents a linear transformation

, defined by .

Example Let be two linear transformations whose matrix representations are respectively

and .



Find the matrix representation of

Solution

Example If for any , then is said to be the matrix representation of

the transformation which transforms to .

Find the matrix representation of

(a) the transformation which transforms any point to ,

(b) the transformation which transforms any point to

Solution



Example It is given that the matrix representing the reflection in the line is

Let be the reflection in the line .

(a) Find the matrix representation of .

(b) The point is transformed by to another point . Find , .

(c) The point is reflected in the line to another point .

Find and .

Solution


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