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.
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.
Alice and Bob have attached ropes to a face of the block of ice and are pulling it in different directions, see Figure 4.34. 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 :
So
Therefore the angle between the vectors is 64.4° (to 1 d.p.).
Note: A tutorial clip of this example is provided on the module website.
Find, to the nearest degree, the angle between the vectors
In this chapter we developed techniques that make it easier to work with vectors. Instead of working with vectors geometrically, it is much more efficient to work with them in component form. When we represent vectors according to their components, engineering problems involving vectors can be solved by carrying out standard algebraic operations.
In the final chapter of Part 2, we will explore how we can use component vectors to solve problems involving forces acting on static objects.
OpenLearn - Part 2: Chapter 4 Applications of vectors Except for third party materials and otherwise, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence, full copyright detail can be found in the acknowledgements section. Please see full copyright statement for details.