We have two distinct ways of calculating the scalar product of two vectors. From Key Point 9 whilst from Key Point 13 . Both methods of calculating the scalar product are entirely equivalent and will always give the same value for the scalar product. We can exploit this correspondence to find the angle between two vectors. The following example illustrates the procedure to be followed.
Find the angle between the vectors and .
The scalar product of these two vectors has already been found in Example 12 to be . The modulus of is . The modulus of is . Substituting these values for and into the formula for the scalar product we find
In general, the angle between two vectors can be found from the following formula:
The angle between vectors , is such that:
- If and find and verify that .
- Find the angle between and .
- Use the definition of the scalar product to show that if two vectors are perpendicular, their scalar product is zero.
- If and are perpendicular, simplify .
- If and , find .
- Show that the vectors and are perpendicular.
- The work done by a force in moving a body through a displacement is given by . Find the work done by the force if it causes a body to move from the point with coordinates to the point .
- Find the angle between the vectors and .
- This follows from the fact that since .
- This follows from the scalar product being zero.
- 39 units.