4.4 Finding the angle between two vectors
The scalar product of two vectors has an important application in calculating the angle between two vectors. If we start with the definition of the scalar product in terms of the magnitudes and directions of the vectors, and rearrange it, then we get the following result.
Angle between two vectors
The angle between any two non-zero vectors and is given by
We can use this result to find the angle between two vectors in component form.
Example 8 Calculating the angle between two vectors in component form
Alice and Bob have attached ropes to a face of the block of ice and are pulling it in different directions, see Figure 34. Vector describes the force applied by Alice, and in component form is given by . Vector describes the force applied by Bob, and in component form is given by .
What is the angle between these vectors, to one decimal place?
First let’s use the components of and to find , and . We have
Using these we can calculate :
Therefore the angle between the vectors is 64.4° (to 1 d.p.).
Find, to the nearest degree, the angle between the vectors