Figure 1 Coordinate system

Solution: (a) Summing the corresponding components of vectors a and b we find:

Figure 2.27

This is the sum of the two vectors is unit–vector form.
(b) Using our results from (a), we can find x-component, and y-component the magnitude of R is

and if R = A + B points in a direction  as measured from the positive x axis, then the tangent of  is found from

If we naively take the arctangent using a calculator, we are told:

which is not correct because (as shown in Fig. 2.27), with c_x negative, and c_y positive, the correct angle must be in the second quadrant. The calculator was fooled because angles which differ by multiples of 180⁰ have the same tangent. The direction we really want is



Example 8: If a − b = 2c, a + b = 4c and c = 3i + 4j, then what are a and b?
Solution: We notice that if we add the first two relations together, the vector b will cancel:
(a − b) + (a + b) = (2c) + (4c)
which gives:

and we can use the last of the given equations to substitute for c; we get

Then we can rearrange the first of the equations to solve for b:

So we have found:
and
Example 9: Given the two displacement

Find the magnitude of the displacement

By using Pythagoras theorem





Multiplying Vectors
Scalar products
There are two ways to “multiply” two vectors together. The scalar product of two vectors and is denoted by . Because of this notation, the scalar product is also called the dot product. Although and are vectors, the quantity is a scalar.

Figure 2.28a

Figure 2.29b

We are given two vectors and , we place these vectors as shown in Figure 2.28a. There is an angle between two vectors. The scalar product (or dot product) of the vectors and is given by

where A is the magnitude of , is the component of in the direction of , Figure 2.28b.
The scalar product is commutative:

The scalar product obeys the commutative law of multiplication; the order of the two vectors does not matter.