Vector Operations
Addition gets two vectors talking, but the everyday work of vector math needs three more moves: scaling a vector, subtracting one from another, and the dot product — which secretly measures how much two vectors point the same way.
Step through each on the grid below.
Scaling and subtraction
Scalar multiplication stretches a vector without turning it: 2a is twice as long
and points the same way; a negative scalar flips it around. It's how you change a
vector's length while keeping (or reversing) its direction.
Subtraction is addition's mirror: a − b is the vector that takes you from b to
a. Geometrically it's the arrow joining the two tips; algebraically you just subtract
componentwise, (3, 1) − (1, 2) = (2, −1). This is the workhorse for "difference
between two points" — displacement, direction-to-target, edges of a shape.
The dot product
The dot product a · b = a₁b₁ + a₂b₂ collapses two vectors into a single number,
and that number has a clean geometric meaning: a · b = |a| |b| cos θ, where θ is the
angle between them. So the dot product is largest when the vectors point the same way,
zero when they're perpendicular, and negative when they oppose.
The animation shows the other reading: drop b straight onto the line of a, and the
shadow it casts is the projection, with length a · b / |a|. Projection is how you
answer "how much of b goes in the direction of a?" — the basis of lighting in
graphics, of components in physics, and of least-squares in data.
Sources
- Lay, D. C., Lay, S. R., & McDonald, J. J. (2021). Linear Algebra and Its Applications (6th ed.). Pearson. — Vector operations, dot product, projections.
- Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press. — Geometry of the dot product.