Definition:Angle between Vectors
Definition
Let $\mathbf v, \mathbf w$ be two non-zero vectors in $\R^n$.
Definition $1$
Let $OV$ and $OW$ be directed line segments representing $\mathbf v$ and $\mathbf w$ respectively.
Let $\theta = \angle VOW$ denote the angle between $OV$ and $OW$ such that:
- $0 \le \theta \le \pi$
that is:
- $0 \degrees \le \theta \le 180 \degrees$
Then $\theta$ is the angle between $\mathbf v$ and $\mathbf w$.
Definition $2$
- Case 1
Suppose that $\mathbf v$ and $\mathbf w$ are not scalar multiples of each other:
- $\neg \exists \lambda \in \R: \mathbf v = \lambda \mathbf w$
Then the angle between $\mathbf v$ and $\mathbf w$ is defined as follows:
Describe a triangle with lengths corresponding to:
- $\norm {\mathbf v}, \norm {\mathbf w}, \norm {\mathbf v - \mathbf w}$
where $\norm {\, \cdot \,}$ denotes vector length:
The angle formed between the two sides with lengths $\norm {\mathbf v}$ and $\norm {\mathbf w}$ is called the angle between vectors $\mathbf v$ and $\mathbf w$.
By convention, the angle is taken between $0$ and $\pi$.
- Case 2
Suppose that $\mathbf v$ and $\mathbf w$ are scalar multiples of each other:
- $\exists \lambda \in \R: \mathbf v = \lambda \mathbf w$
As $\mathbf v$ and $\mathbf w$ as non-zero, $\lambda \ne 0$.
If $\lambda > 0$, then the angle between $\mathbf v$ and $\mathbf w$ is defined as a zero angle, that is:
- $\theta = 0$
If $\lambda < 0$, then the angle between $\mathbf v$ and $\mathbf w$ is defined as a straight angle, that is:
- $\theta = \pi$
If either $\mathbf v$ or $\mathbf w$ is zero, the angle between $\mathbf v$ and $\mathbf w$ is not defined.
Also note that in all cases:
- $0 \le \theta \le \pi$
Also see
- Angle Between Non-Zero Vectors Always Defined
- Cosine Formula for Dot Product
- Angle Between Vectors in Terms of Dot Product
- Results about angles between vectors can be found here.
Sources
- For a video presentation of the contents of this page, visit the Khan Academy.