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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?

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    $\begingroup$ If the dot product of two vectors is $0$, they are orthogonal, which means perpendicular. $\endgroup$ – Andy Bromberg May 26 '13 at 22:53
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    $\begingroup$ Yes: $a \perp b \iff \langle a, b \rangle = 0$ $\endgroup$ – Ayman Hourieh May 26 '13 at 22:54
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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 $\vec{u}\cdot\vec{v} = \sum_{i=1}^n\: a_i b_i.$

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This is a numerical answer since explanatory answers have already been given.

$\vec a = (1,-2,3)$ 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.

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  • $\begingroup$ Is it true to say each one vector is in the null space of the combined space of the other two? $\endgroup$ – Michael Barton Nov 14 '17 at 22:03
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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 $\vec{w}$.

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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 $90^{\circ}$.
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