Sum and difference of vectors

The sum of any two or more vectors is called the resultant of the given vectors. The sum of vectors is governed by triangle, parallelogram and polygon laws of vector addition.

(1) Triangle law of vector Addition

Adding two vectors involves joining two vectors such that the initial point of the second vector is the end point of first vector and the resultant is obtained by completing the triangle with the vector whose initial point is the initial point of the first vector and whose end points the end point of the second vector.

From the figure above a + b is the resultant of vectors a and b as shown below

(2) The parallelogram law

When two vectors have a common initial point say P, then their resultant is obtained by completing a parallelogram, where the two vectors are the sides of the diagonal through P and with initial point at P

Example 1

Find the resultant of vectors u and v in the following figure.

Solution

To get the resultant of vectors u and v, you need to complete the parallelogram as shown in the following figure

From the figure above, the result of u and v is PR = U + V = U + V

Note that by parallelogram law of vector addition, commutative property is verified.

Polygon law of vector addition

If you want to add more than two vectors, you join the end point to the initial point of the vectors one after another and the resultant is the vector joining the initial point of the first vector to the end point of the last vector

Example 2

Find the resultant of vectors a, b, c and d as shown in the figure below.

Solution

In the figure above P is the initial point of a, b has been joined to a at point Q and c is joined to b at R, while d is joined to c at point S and PT = a + b + c + d which is the resultant of the four vectors.

Two vectors are said to be opposite to each other if they have the same magnitude but different directions

From the figure above a and b have the same magnitude (3m) but opposite direction.

So a and b are opposite vectors.

Opposite vectors have zero resultant that is if a and b are opposite vectors, then a + b = 0

Example 3

Find the vector p opposite to the vector r = 6i – 2j

Solution

Let $p = ai + bj$

Since $p$ and $r$ are opposite to each other, then $p + r = 0$

So $(ai + bj) + (6i - 2j) = 0$ $(ai + 6i) + (bj - 2j) = 0i + 0j = 0$ $ai + 6i = 0i$...........(1) $bj - 2j = 0j$...........(2)

$a + 6 = 0$ which means $a = -6$ $b - 2 = 0$ meaning that $b = 2$

but $p = ai + bj$ $p = -6i + 2j$

Therefore $p = -6i + 2j$

Normally when subtracting one vector from another the result obtained is the same as that of addition but to the opposite of the other vector.

Therefore the difference of two vectors is also the resultant vector