What is the Scalar (Dot) Product?
The scalar product (also called the dot product) of two vectors and is an operation that takes two vectors as inputs and returns a scalar as output. For and , it is defined as:
The key thing to appreciate straight away: unlike multiplying two scalars or scaling a vector, the dot product collapses two vectors into a single number. This number encodes important geometric information about the relationship between the two vectors — specifically, how much they "point in the same direction".
The notation uses a bold dot (·) between the two vectors. Never write for the scalar product — that is reserved for the vector (cross) product, which gives a vector, not a scalar.
The formula works identically in 2D (just drop the term) and in 3D. In IB HL, you will almost always work in 3D.
Computing the Scalar Product — Worked Examples
Example 1: Let and . Find .
Step 1: Multiply corresponding components.
Step 2: Add the products.
Note that — the same result, illustrating commutativity.
Example 2: Let and . Find .
Notice that the middle term vanishes immediately because . Always check for zeros to speed up your calculation.
Work left to right: multiply each pair, write down the three products, then sum them. Do not try to do it all in one mental step — sign errors are very common under exam pressure.