What Are Orthogonal (Perpendicular) Vectors?

 

Hello. This post will talk about orthogonal (perpendicular) vectors in the n-th dimension \mathbb{R}^{n}. It is assumed that one knows about dot products.


Review of Dot Products and The Cosine of An Angle

Recall that we can find the cosine of an angle \theta using the dot product of vectors \boldsymbol{u} and \boldsymbol{v} in \mathbb{R}^{n} and its norms.

The formula for finding the cosine of an angle \theta is:

\displaystyle \cos(\theta) = \dfrac{\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{v}}{||\boldsymbol{u}|| ||\boldsymbol{v}||}

Applying the inverse cosine function or the arccos function (which is the same thing) to both sides of the equations, we can isolate the angle \theta.

\displaystyle \theta = \cos^{-1}(\dfrac{\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{v}}{||\boldsymbol{u}|| ||\boldsymbol{v}||})

or

\displaystyle \theta = \arccos(\dfrac{\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{v}}{||\boldsymbol{u}|| ||\boldsymbol{v}||})

where \theta can take values from 0 to \pi (inclusive).

Orthogonal Vectors

From the above, if \boldsymbol{u} \cdot \boldsymbol{v} = 0 then \theta is a right angle (\theta = 90 degrees or \dfrac{\pi}{2} in radians). We can also say that the vectors \boldsymbol{u} and \boldsymbol{v} or perpendicular or orthogonal in \mathbb{R}^{n}.

The zero vector in \mathbb{R}^{n} is orthogonal to every vector in \mathbb{R}^{n}.

Proof

Suppose we have the zero vector \boldsymbol{0} and another arbitrary vector such as (a_{1}, a_{2}, \dots , a_{n}) in \mathbb{R}^{n}. Taking the dot product of these two vectors gives us:

\displaystyle \begin{array} {lcl} \boldsymbol{0} \boldsymbol{\cdot} (a_{1}, a_{2}, \dots , a_{n}) & = & (0, 0, 0, \dots, 0) \boldsymbol{\cdot} (a_{1}, a_{2}, \dots , a_{n}) \\ & = & 0 \cdot a_1 + 0 \cdot a_2 + \dots + 0 \cdot a_n \\ & = & 0 + 0 + \dots + 0 \\ & = & 0 \end{array}


Examples

We illustrate the concept of orthogonal vectors with a few examples.

Example One

In \mathbb{R}^{2} the dot product of the vectors \boldsymbol{a} = (2, -9) and \boldsymbol{b} = (8, 1) is 7. These two vectors are not orthogonal to one another.

\displaystyle \begin{array} {lcl} \boldsymbol{a} \boldsymbol{\cdot} \boldsymbol{b} & = & (2, -9) \boldsymbol{\cdot} (8, 1) \\ & = & 2 \times 8 + (- 9) \times 1 \\ & = & 16 - 9\\ & = & 7\\ \end{array}

Example Two

Consider two vectors in \boldsymbol{c} = (-19, 1, 0, 2) and \boldsymbol{d} = (0, 20, 32, -10) in {\mathbb{R}^{4}. Are these two vectors orthogonal (perpendicular) in \mathbb{R}^{4}?

Answer

To determine whether the two vectors are orthogonal (perpendicular) in \mathbb{R}^{4}, we compute the dot product of vectors \boldsymbol{c} and \boldsymbol{d}.

\displaystyle \begin{array} {lcl} \boldsymbol{c} \boldsymbol{\cdot} \boldsymbol{d} & = & (-19, 1, 0, 2) \boldsymbol{\cdot} (0, 20, 32, -10) \\ & = & -19 \times 0 + 1 \times 20 + 0 \times 32 + 2 \times (-10)\\ & = & 0 + 20 + 0 - 20\\ & = & 0\\ \end{array}

The vectors \boldsymbol{c} and \boldsymbol{d} are orthogonal to one another in \mathbb{R}^{4}.


Reference

Elementary Linear Algebra (Tenth Edition) by Howard Anton.

The featured image is from https://www.math.hmc.edu/calculus/tutorials/vectoranalysis/gif/vector9.gif.

One thought on “What Are Orthogonal (Perpendicular) Vectors?

  1. Pingback: Relating Dot Products, Orthogonal Vectors and Cross Products – DK Math Stats Blog

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