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How do dot products show whether two vectors are perpendicular?

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vector parallelogram for addition and subtraction — linear algebra One method of adding and subtracting vectors is to place their tails together and then supply two more sides to form a parallelogram. The vector from their tails to the opposite corner of the parallelogram is equal to the sum of the original vectors. The vector between their heads (starting from the vector being subtracted) is equal to their difference.

In linear algebra, the dot product of two vectors, $𝐚$ and $𝐛$ , is defined as:

$𝐚 \cdot 𝐛 = | 𝐚 | | 𝐛 | \cos θ$

where $| 𝐚 |$ and $| 𝐛 |$ are the magnitudes of the vectors and $θ$ is the angle between them.12

Because the cosine of a 90-degree angle (perpendicular) is zero, the dot product of two non-zero vectors is exactly zero if and only if the vectors are perpendicular. In this context, perpendicularity is often referred to as orthogonality. Therefore, calculating the dot product provides a straightforward algebraic method to determine if two vectors are at right angles without needing to measure the angle directly.12