A is multiplied could be a scalar having its own physical dimension. Then, the dimension of λ A is the product of the dimensions of λ and A . For example, if we multiply a constant velocity vector by duration (of time), we get a displacement vector. .
ADDITION AND SUBTRACTION OF VECTORS — GRAPHICAL METHOD As mentioned in section . , vectors, by definition, obey the triangle law or equivalently, the parallelogram law of addition. We shall now describe this law of addition using the graphical method. Let us consider two vectors A and B that lie in a plane as shown in Fig.
. (a). The lengths of the line segments representing these vectors are proportional to the magnitude of the vectors. To find the sum A + B , we place vector B so that its tail is at the head of the vector A , as in Fig.
. (b). Then, we join the tail of A to the head of B . This line O Q represents a vector R, that is, the sum of the vectors A and B .
Since, in this procedure of vector addition, vectors are Fig. . (a) Two equal vectors A and B . (b) Two vectors A ′ and B ′ are unequal though they are of the same length .
Fig. . (a) Vector A and the resultant vector after multiplying A by a positive number . (b) Vector A and resultant vectors after multiplying it by a negative number – and – .
. (c) (d) Fig. . (a) Vectors A and B .
(b) Vectors A and B added graphically. (c) Vectors B and A added graphically. (d) Illustrating the associative law of vector addition. arranged head to tail, this graphical method is called the head-to-tail method .
The two vectors and their resultant form three sides of a triangle, so this method is also known as triangle method of vector addition . If we find the resultant of B + A as in Fig. . (c), the same vector R is obtained.
Thus, vector addition is commutative