Orthogonal vectors are vectors that are perpendicular to each other. This means they form a 90-degree angle where they meet. In this example, vector a points horizontally to the right, while vector b points vertically upward. These two vectors are orthogonal because they meet at a right angle.
The mathematical test for orthogonality is the dot product. Two vectors are orthogonal if and only if their dot product equals zero. For example, vector a equals 3, 0 and vector b equals 0, 2. Their dot product is 3 times 0 plus 0 times 2, which equals zero. This confirms that these vectors are orthogonal.
Here we see an example of non-orthogonal vectors. Vector c equals 2, 0 and vector d equals 1, 2. These vectors do not form a 90-degree angle. Their dot product is 2 times 1 plus 0 times 2, which equals 2, not zero. Since the dot product is not zero, these vectors are not orthogonal.
Orthogonality extends naturally to three dimensions. Here we see the standard basis vectors i, j, and k. Vector i points along the x-axis, vector j along the y-axis, and vector k along the z-axis. These three vectors are mutually orthogonal, meaning each pair forms a 90-degree angle. This forms the foundation of our three-dimensional coordinate system.
Orthogonal vectors have numerous practical applications. They are fundamental in computer graphics for 3D modeling, in physics for decomposing forces, in machine learning for data analysis, and in engineering for signal processing. Remember, two vectors are orthogonal if and only if their dot product equals zero. This simple mathematical relationship has profound implications across many fields of science and technology.