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 ?

Answer

To determine whether the two vectors are orthogonal (perpendicular) in , we compute the dot product of vectors and .

The vectors and are orthogonal to one another in .

__Reference__

Elementary Linear Algebra (Tenth Edition) by Howard Anton.

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