# which vectors are perpendicular to each other?

which vectors are perpendicular to each other?

$\vec a = (1, -2, 3)$, $\vec b = (5, 4, 1)$, $\vec c = (1, 0, -5)$

Do i just take the dot product of 2 of them. If the dot product they are at $90^\circ$? But how do i know if there perpendicular?

• If the dot product of two vectors is $0$, they are orthogonal, which means perpendicular. – Andy Bromberg May 26 '13 at 22:53
• Yes: $a \perp b \iff \langle a, b \rangle = 0$ – Ayman Hourieh May 26 '13 at 22:54

If the dot product two vectors is $$0$$, they are orthogonal; in other words, they are perpendicular.

The dot product between two vectors $$\vec u, \vec v$$ is given by $$\vec{u}\cdot\vec{v} = |\vec{u}||\vec{v}|\cos(\theta)$$, so $$\vec u \cdot \vec v = 0 \implies \cos \theta = 0 \implies \theta = \pi/2 \;\;(90^\circ).$$

(Recall: two vectors that are orthogonal (perpendicular) form a right angle $$\theta = \pi/2 = 90^\circ$$.)

Algebraic definition would be $$\vec{u}\cdot\vec{v} = \sum_{i=1}^n\: a_i b_i.$$

$\vec a = (1,-2,3)$ and $\vec b=(5,4,1)$ are perpendicular (orthogonal)

$\vec b = (5,4,1)$ and $\vec c=(1,0,-5)$ are also perpendicular.

Why?

$$\vec a\cdot \vec b=\begin{bmatrix}1\\-2\\3\end{bmatrix}\cdot\begin{bmatrix}5\\4\\1\end{bmatrix}=1\cdot 5-2\cdot 4+3\cdot 1=5-8+3=0$$

$$\vec b\cdot \vec c=\begin{bmatrix}5\\4\\1\end{bmatrix}\cdot\begin{bmatrix}1\\0\\-5\end{bmatrix}=5\cdot 1+4\cdot 0+1\cdot -5=5-0-5=0$$

This is because their scalar products are both zero.

• Is it true to say each one vector is in the null space of the combined space of the other two? – Michael Barton Nov 14 '17 at 22:03

Two vectors are perpendicular if the angle between them is $\frac{\pi}{2}$, i.e., if the dot product is $0$. This follows from the fact that for two vectors $\vec{v}, \vec{w}$, we have $\vec{v}\cdot\vec{w} = |\vec{v}||\vec{w}|\cos(\theta)$, where $\theta$ is the angle between $\vec{v}$ and $\vec{w}$.

To know the vectors that are perpendicular or orthogonal to each other, the following must be taken into consideration:

1. the scalar (dot) product between the two considered vectors must be zero $$(0)$$.
2. the angle between the two considered vectors must be $$90^{\circ}$$.