Parallel and Orthogonal Vectors

Two vectors are parallel if one of the vectors is a scalar multiple of the other. The following vectors are parallel:

v1 = < 2, 3, 4 >  v2 = < 6, 9, 12 >

If you multiply vector v1 by 3, you get vector v2, which makes the two vectors parallel.

Orthogonal means perpendicular. Two vectors are orthogonal if their dot product is 0. Use the cross product to calculate an orthogonal 3D vector. Calculating an orthogonal vector for a 2D vector is easier. If you have a 2D vector v

v = < x, y >

There are two vectors orthogonal to v.

u1 = < y, -x >  u2 = < -y, x >

Two vectors are orthonormal if they’re both unit vectors (have length 1) and are orthogonal.


As I finish I want you to understand the terms norm, normal, normalize, and orthonormal. Norm is the vector’s length. A normal is a vector that is perpendicular to a surface. Normalizing a vector converts it to a unit vector, a vector of length 1. Two vectors are orthonormal if they are unit vectors and are perpendicular to each other.

There is source code to accompany this article. The source code consists of C++ classes for 2D and 3D vectors. If you have any questions or comments about this article, send them to mark-AT-meandmark-DOT-com.

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