Definition:Angle Between Vectors

Definition

Let $\mathbf v, \mathbf w$ be two non-zero vectors in $\R^n$.

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:

$\theta = \pi$

Comment

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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

Sources

• For a video presentation of the contents of this page, visit the Khan Academy.