AHL 3.13 (HL)—Scalar product

The scalar product tests angles and perpendicularity.

a·b=|a||b|cosθ converts two vectors into a number, so its sign and value reveal the angle between them.

Example

(2,1,0)·(1,−2,0)=2−2=0, so the vectors are perpendicular.

Use components for calculation, then interpret zero, positive or negative as right, acute or obtuse angle evidence.

The scalar product is commutative and produces a scalar; it is not the vector product and cannot give a direction.

For non-zero vectors $\mathbf a=(a_1,a_2,a_3)$ and $\mathbf b=(b_1,b_2,b_3)$, $\mathbf a\cdot\mathbf b=a_1b_1+a_2b_2+a_3b_3$ and $\cos\theta=(\mathbf a\cdot\mathbf b)/(|\mathbf a||\mathbf b|)$. Zero dot product is equivalent to perpendicularity; parallel vectors satisfy $|\mathbf a\cdot\mathbf b|=|\mathbf a||\mathbf b|$, with the sign distinguishing the same and opposite directions.