Scalars and Vectors

Scalars have magnitude only. Temperature, speed, mass, and volume are examples of scalars.

Vectors have magnitude and direction. The magnitude of $\vec{v}$ is written |$\vec{v}$| $\equiv$ v . Position, displacement, velocity, acceleration and force are examples of vector quantities. Vectors have the following properties:

1.

Vectors are equal if they have the same magnitude and direction.

2.

Vectors must have the same units in order for them to be added or subtracted.

3.

The negative of a vector has the same magnitude but opposite direction.

4.

Subtraction of a vector is defined by adding a negative vector:

$$\vec{A}$$ - $$\vec{B}$$ = $$\vec{A}$$ + (- $$\vec{B}$$)

5.

Multiplication or division of a vector by a scalar results in a vector for which

(a)

only the magnitude changes if the scalar is positive

(b)

the magnitude changes and the direction is reversed if the scalar is negative.

6.

The projections of a vector along the axes of a rectangular co-ordinate system are called the components of the vector. The components of a vector completely define the vector.

 

Figure 3.1: Projections of a vector in 2-D.

$$\theta {\frac{A_{x}}{A}} \Rightarrow \theta$$ =

$$\theta {\frac{A_{y}}{A}} \Rightarrow \theta$$ =

We can invert these equations to find A and $\theta$ as functions of A_x and A_y . By Pythagoras we have,

$$\sqrt{A_{x}^{2} + A_{y}^{2}}$$

and from the diagram,

$$\theta {\frac{A_{y}}{A_{x}}}$$ =

$$\theta \tan^{-1}_{} {\frac{A_{y}}{A_{x}}}$$ =

7.

To add vectors by components: $\vec{R}$ = $\vec{A}$ + $\vec{B}$ + $\vec{C}$ +...

(a)

Find the components of all vectors to be added.

(b)

Add all x components to get R_x = A_x + B_x + C_x + ...
Add all y components to get R_y = A_y + B_y + C_y +...

(c)

Then

$$\vec{R} \sqrt{R_{x}^{2} + R_{y}^{2}}$$ =

$$\theta \tan^{-1}_{} {\frac{R_{y}}{R_{x}}}$$ =