Vector Addition

Vectors can be added using the parallelogram rule or the ‘nose-to-tail’ method.

Two vectors a and b represented by the line segments can be added by joining the ‘tail’ of vector b to the ‘nose’ of vector a. Alternatively, the ‘tail’ of vector a can be joined to the ‘nose’ of vector b.

Example:

Find the sum of the two given vectors a and b.

Solution:

Draw the vector a. Draw the ‘tail’ of vector b joined to the ‘nose’ of vector a. The vector a + b is from the ‘tail’ of a to the ‘nose’ of b.

Example:

Given that , find the sum of the vectors.

Solution:

In vector addition, the intermediate letters must be the same. Since PQR forms a triangle, the rule is also called the triangle law of vector addition.

Vector Addition is Commutative

We will find that vector addition is commutative, that is a + b = b + a

This can be illustrated in the following diagram.

Vector Addition is Associative

We also find that vector addition is associative, that is (u + v) + w = u + (v + w ).

This can be illustrated in the following two diagrams. Notice that (u + v) + w and u + (v + w ) have the same magnitude and direction and so they are equal.

Example:

ABCD is a quadrilateral. Simplify the following:

The following video shows another example of vector addition.

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