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Dot product
Intermediate
Scalar product measuring how much two vectors align.
A·B = AB cosθThe dot (scalar) product A·B = AB cosθ = AxBx + AyBy + AzBz gives a scalar. It measures how much one vector points along the other, and is zero for perpendicular vectors. Work is a dot product: W = F·d.
A force (2, 3) N acting over a displacement (4, 0) m does W = 8 J.
The result is a number, not a vector, and the order does not matter.
In more detail
The dot product of two vectors is A·B = AB cos θ, a scalar that measures how much one vector points along another. It is zero when the vectors are perpendicular.
Real-world uses
- Work is the dot product of force and displacement, so only the component of force along the motion does work.
- Computer graphics use dot products to calculate lighting angles on surfaces.
- The work done by a constant force on a sliding box is the dot product of the force and the displacement, so only the component along the motion counts.
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Frequently asked questions
Why is work the dot product?
Only the component of force in the direction of displacement contributes, which is exactly what the dot product selects.
What is the dot product of perpendicular vectors?
Zero, because cos 90 degrees is zero.
Why is the dot product zero for perpendicular vectors?
Because cos 90° is zero, so one vector has no component along the other.
Related topics in Measurement & Math Tools
Parallelogram law · Vector components · Unit vectors · Cross product · Triple products · Calculus: derivatives