Scalar (Dot) Product of Vectors
Simple Explanation
The scalar (dot) product multiplies two vectors together to produce a single number (a scalar), not a new vector. It measures how much one vector "lines up" with the direction of another.
Why Do We Need It?
The dot product is how physics defines quantities like work — work is only done by the part of a force that acts along the direction of motion, and the dot product captures exactly that.
Formula
Scalar (Dot) Product
A · B = AB cosθ
The scalar product of two vectors gives a single number (scalar), representing how much one vector points in the direction of the other.
- A · B
- — The scalar (dot) product, a number with no direction
- A, B
- — Magnitudes of the two vectors
- θ
- — The angle between the two vectors
When to use it: Use when combining two vectors should produce a plain number, such as calculating work done (force · displacement).
Worked Example
Calculating a scalar product
A force of 10 N acts at 60° to the direction of a 5 m displacement. Find the scalar product (which equals the work done).
Why Does This Work?
When θ = 0° (vectors fully aligned), cosθ = 1, so the dot product simply becomes the product of the magnitudes — its maximum possible value. When θ = 90° (perpendicular vectors), cosθ = 0, so the dot product is zero — a force perpendicular to motion does no work, exactly matching physical intuition.
Real-Life Example
Carrying a bag while walking
A person carries a bag by holding it vertically while walking horizontally.
The force holding the bag up is perpendicular (90°) to the horizontal walking direction — since cos90° = 0, the dot product (work done by that lifting force on the horizontal motion) is zero, matching the physical fact that holding a bag steady does no work on its horizontal motion.
Practice
Two vectors of magnitude 8 and 6 act at 0° to each other (fully aligned). Find their scalar product.
MediumCommon mistake
Expecting the dot product to give a vector — it always produces a plain NUMBER (scalar), with no direction, unlike the vector (cross) product.
Quick Review
- A · B = AB cosθ produces a scalar (number), not a vector.
- Maximum when vectors are aligned (θ=0°); zero when perpendicular (θ=90°).
- Used to define quantities like work done by a force.