It was mentioned earlier that, displacement vectors are added to displacement vectors, or velocity vectors are added to velocity vectors. Just as it is meaningless to add scalar quantities of different kinds, such as mass and temperature, so also it is meaningless to add vector quantities of different kinds, such as displacement and electric field strength. However, like scalars, vectors of different kinds can be multiplied by one another to generate quantities of new physical dimensions. Because vectors have direction as well as magnitude, vector multiplication cannot follow exactly the same rules as the algebraic rules of scalar multiplication. There are new rules of multiplication for vectors. It will be useful to define three kinds of multiplication operations for vectors:
- Multiplication of a vector by a scalar,
- Multiplication of two vectors in such a way so as to yield a scalar.
- Multiplication of two vectors in such a way so as to yield another vector.
The first case, that is, multiplication of a vector by a scalar has been dealt with in the section; multiplication of vectors by real numbers. Hence, we move on to the second case, that is, multiplication of two vectors in such a way as to yield a scalar. Such an operation is known as the scalar or dot product.
the cosine of the angle q between the two vectors.
It is important to remember that there are two different angles between a pair of vectors, depending on the direction of rotation. However, only the smaller of the two is taken in vector multiplication.
Since, a and b are scalars and cos q is a pure number, the scalar product of two vectors is a scalar.
As shown in the figure, the scalar product of two vectors can be regarded as the product of the magnitude of one vector and the component of the other vector in the direction of the first.
component of a in the direction of the unit vector.
For example,
