Consider the two vectors and shown in Figure 29.
Figure 29 :
Note that the tails of the two vectors coincide and that the angle between the vectors is labelled . Their scalar product, denoted by , is defined as the product . It is very important to use the dot in the formula. The dot is the specific symbol for the scalar product, and is the reason why the scalar product is also known as the dot product . You should not use a sign in this context because this sign is reserved for the vector product which is quite different.
The angle is always chosen to lie between 0 and , and the tails of the two vectors must coincide. Figure 30 shows two incorrect ways of measuring .
Figure 30 :
The scalar product of and is:
“The modulus of the first vector, multiplied by the modulus of the second vector,
multiplied by the cosine of the angle between them.”
Clearly and so
Thus we can evaluate a scalar product in any order: the operation is commutative .
Vectors and are shown in the Figure 31. Vector has modulus 6 and vector has modulus 7 and the angle between them is . Calculate .
The angle between the two vectors is . Hence
The scalar product of and is 21. Note that a scalar product is always a scalar.
Find where is the unit vector in the direction of the positive axis.
Because is a unit vector its modulus is 1. Also, the angle between and itself is zero. Therefore
So the scalar product of with itself equals 1. It is easy to verify that and .
Find where and are unit vectors in the directions of the and axes.
Because and are unit vectors they each have a modulus of 1. The angle between the two vectors is . Therefore
That is .
The following results are easily verified:
Generally, whenever any two vectors are perpendicular to each other their scalar product is zero because the angle between the vectors is and .
The scalar product of perpendicular vectors is zero.