Question

How do I find the angle between two vectors in three-dimensional space?

Answer 1

This problem deals with finding the angle between two vectors in a three-dimensional space. This is done with the help of the dot product of two vectors. Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates.

Complete step-by-step solution:
Let the two vectors be $\overrightarrow a$ and $\overrightarrow b$.
Now let the angle between these two vectors $\overrightarrow a$ and $\overrightarrow b$ be $\theta$.
From the dot product, the dot product of two vectors is given by the products of the magnitudes of the two vectors and the cosine of the angle between these two vectors.
The dot product of two vectors which is described above, is mathematically expressed below:
$\Rightarrow \overrightarrow a \cdot \overrightarrow b = \left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|\cos \theta$
From here the cosine of the angle $\theta$ is obtained, which is shown below:
$\Rightarrow \dfrac{{\overrightarrow a \cdot \overrightarrow b }}{{\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|}} = \cos \theta$
Now rearranging the above equation, as given below:
$\Rightarrow \cos \theta = \dfrac{{\overrightarrow a \cdot \overrightarrow b }}{{\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|}}$
Now the angle $\theta$, is found by applying the cosine inverse on both sides of the equation as given below:
$\Rightarrow {\cos ^{ - 1}}\left( {\cos \theta } \right) = {\cos ^{ - 1}}\left( {\dfrac{{\overrightarrow a \cdot \overrightarrow b }}{{\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|}}} \right)$
Thus the expression of the angle $\theta$ is given below:
$\Rightarrow \theta = {\cos ^{ - 1}}\left( {\dfrac{{\overrightarrow a \cdot \overrightarrow b }}{{\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|}}} \right)$

The angle between two vectors in three-dimensional space is $\theta = {\cos ^{ - 1}}\left( {\dfrac{{\overrightarrow a \cdot \overrightarrow b }}{{\left| {\overrightarrow a } \right|\left| {\overrightarrow b } \right|}}} \right)$

Note: In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used.