Vectors

2 types of physical quantities

• In Physics not only is ‘how much’ important but also in what direction. Therefore in physics we have vector quantities.

1. Scalar – magnitude ( how much) only

2. Vector – magnitude & direction

Vector & scalar cont.

• Most quantities with which we are familiar with are scalar quantities.

• In print as already indicated both your book & myself will indicate vector in bold print & scalar in normal text.

• Vector quantities are distinguished from scalar quantities by a small arrow above the symbol representing the quantity.

v v

vector scalar

Quantities

Scalar • Time (t)• Mass (m)• Distance (d)• Speed (v)• Accelaration (a)• Energy (E)

• Vector

• Displacement (d)• Velocity (v)• Acceleration (a)• Force (F)• Momentum (P)

• A scalar quantity is always the magnitude of the corresponding vector quantity.

• Ex.

• d is the magnitude of d.

• v is the magnitude of v.

• Vectors define mag. & dir. & are independent of the origin.

• Therefore, if 2 vectors have the same mag. & dir. they are the same vector.

ab

a & b are the same vector.

Arithmetic of vectors

• Terminology:

tail

tip

Addition – consider the vectors a & b

a) Put vectors end to end as presented in steps b) through d)

b) Draw vector a.c) Put the tail of vector b at the tip of vector a.

d) Draw the resultant vector – from the tail of a to the tip of b. That is, from the origin to the destination.

a b

b) Draw vector a

a

c) Put the tail of vector b at the tip of vector a.

a b

d) Draw the resultant vector – from the tail of a to the tip of b.

a b

a + b

Does the commutative law apply to vectors (that is does a+b = b+a)?

a + b

b a

b + a

Clearly, a+b = b+a. Inductively, the commutative law appears to be true.

Does the associative law apply to vectors?

i.e. does (a+b) + c = a + (b+c)?

Consider the below vectors

a b c

ab

a+b

a+b c

(a+b) + c

bc

b+c b+c a

(b+c) + a

(a+b) + c = (b+c) + a. Therefore, inductively, the associative law appears to be true.

Notice, one must first draw the quantity (a+b), then transfer it.

Vector subtraction: a - b

a – b = a + (-b)

a) Draw –b (-b, has the same magnitude b with 180° opposite direction).

b) Then add a + (-b) using the rules of vector addition.

ab

b-b

a

-b(a-b)

Vector multiplication: i.e. multipli- cation of a vector x a scalar quantity.

Ex. 5a. a

Draw 5 a vectors tail to tip.

Direction will always be maintained in vectorMultiplication.

Vector x a scalar always = a vector in the same direction. (Ding, Ding, Ding!!!!!)

5aa

Vector division: i.e. dividing a vector by a scalar.

• Simply reduce the length of a vector the appropriate # of times.

• Ex. a/3 = a vector 1/3 the length of vector a & in the same direction.

a

a/3

Direction

• N,E,R,up are typically designated as + direction by convention. S,W,L,down are – direction. There are times when it is convenient to break w/ that convention, however.

• In doing problems we will only use +/- when we cannot more explicitly give the direction. Ex., 5 m E is better than +5 m.

• We will almost always measure angles from the nearest x axis. There are times when we have problems that give us little choice, however. We will give angles relative to the x axis thus NE, SW, above horizontal etc.. Your book does not tend to do that but in the world it is typically done that way. For ex., the weatherman will give wind direction as NW not WN.

Vector Definitions

• Components = rectangular form of a vector.

• Resultant = the result of adding components. The answer will initially be in rectangular form. It will be necessary to convert to the polar form.

Adding vectors by component addition

1. Diagram problem – vector diagram not a picture diagram. You may choose to make a picture diagram also for you own purposes.

2. Resolve each vector into its components (show work!!!)

3. Add all x components

Add all y components

4. Find the resultant vector from the sum of the components (show work!!!)

Summary of what is required when working w/ vectors

1. We will not use the Pythagorean theorem. We will work all problems using trig.. Your book & I disagree on this. Rationale: I want you to learn how to find the trig. function solution to problems. There is always a trig. function way.

2. When doing trig. start w/ the definition of a trig. function as it relates to the triangle in question. Ex. Sin = vy/v not o/h.

3. Do algebra before plugging in #s.

4. Always show an x,y table when adding vectors that are not in the same or 1800 opposite direction. Do not show a table when adding vectors in same or 1800 opp. Take care of that mathematically w/ signs.

5. Show trig. only when specifically assigned. Use P-> R & R->P the remainder of the time. Indicate how you are arriving @ the answer on the x,y table by writing to the right of the table by the approp, vector either “trig.” or “P-> R” or “R->P”.

6. On odd #ed problems show the trig. for resolving 1 vector. Use P->R function on your calculator to resolve the other vectors.

7. On even #ed problems show the trig. for the resultant vector. On odd #ed problems you may determine the resultant vector using P->R on your calculator.

8. It is not necessary to show trig. or R -> P on N,E,S,W,R,L,up,down vectors. Just record in the x,y table. Ex. 5 m S = 0, -5 in x/y table.