Figure 1 Coordinate system

Calculating the Scalar Product Using Components
We should calculate the scalar product by using components. Let's first work out the scalar products of the unit vectors. This is easy, since i, j, and k all have magnitude 1 and are perpendicular to each other.


Now we express and in terms of their components, expend the product, and use these products of unit vectors:







Here some scalar products are equal to zero according to Eqs.8. That is why we get

If two vectors are perpendicular then their scalar product is zero.
Example 10: Vector A extends from the origin to a point having polar coordinates (7, 70⁰) and vector B extends from the origin to a point having polar coordinates (4, 130⁰).
Find A · B.
We can use Eq. 2.15 to find We have the magnitudes of the two vectors (namely A = 7 and B = 4) and the angle ϕ between the two is

Then we get:

To use the second approach, we first need to find the components of the two vectors. Since the angles of and are given with respect to the + x-axis, and these angles are measured in the sense from the + x-axis to the +y-axis, we can find,






The z-components are zero because both vectors lie in the xy- plane. From Eq. (9) the scalar product is



Example 11: Find the angle between A = −5i − 3j + 2k and B = −2j − 2k.
allows us to find the cosine of the angle between two vectors as long as we know their magnitudes and their dot product. The magnitudes of the vectors A and B are:


and their dot product is:

Then from Eq. 1.7, if  is the angle between A and B, we have

which then gives