# What Are Orthogonal (Perpendicular) Vectors?

Hello. This post will talk about orthogonal (perpendicular) vectors in the n-th dimension . 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 using the dot product of vectors and in and its norms.

The formula for finding the cosine of an angle is:

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 .

or

where can take values from 0 to (inclusive).

Orthogonal Vectors

From the above, if then is a right angle ( = 90 degrees or in radians). We can also say that the vectors and or perpendicular or orthogonal in .

The zero vector in is orthogonal to every vector in .

Proof

Suppose we have the zero vector and another arbitrary vector such as () in . Taking the dot product of these two vectors gives us:

Examples

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

Example One

In the dot product of the vectors and is 7. These two vectors are not orthogonal to one another.

Example Two

Consider two vectors in and in . Are these two vectors orthogonal (perpendicular) in ?