The Dot Product of Two Vectors
Simple Explanation
The dot product (or scalar product) combines two vectors into a single number: multiply their corresponding components together, then add the results. a·b = a₁b₁ + a₂b₂ + a₃b₃.
Why Do We Need It?
The dot product is the key tool for finding the angle between two vectors, and for testing whether two vectors are perpendicular (their dot product is exactly zero).
Formula
The Dot Product
a · b = a₁b₁ + a₂b₂ + a₃b₃
The dot (scalar) product of two vectors is found by multiplying corresponding components and adding the results — the output is always a single number (a scalar), not a vector.
- a = (a₁,a₂,a₃), b = (b₁,b₂,b₃)
- — the two vectors being combined
When to use it: Whenever two vectors need to be combined into a single scalar value, especially to find the angle between them.
Worked Example
Find the dot product of two vectors
Find the dot product of a=(1,2,3) and b=(4,−5,6).
Why Does This Work?
The dot product can be shown (using the Law of Cosines applied to the triangle formed by a, b, and a−b) to equal |a||b|cosθ — this is exactly why it is the key ingredient for finding the angle θ between two vectors, and why it equals zero precisely when θ=90° (since cos90°=0).
Real-Life Example
Calculating work done by a force
A force pushes an object in a direction that is not exactly the same as the object's displacement.
The work actually done is the dot product of the force vector and the displacement vector — accounting exactly for how much of the force acts along the direction of motion.
Practice
Find the dot product of a=(2,0,3) and b=(−1,4,2).
MediumCommon mistake
Forgetting that the dot product's result is a single SCALAR (a plain number), not another vector — it should never be written as a triple like the inputs.
Quick Review
- a·b = a₁b₁ + a₂b₂ + a₃b₃.
- The result is always a scalar (single number), never a vector.
- a·b = 0 exactly when a and b are perpendicular.