Scalar product

Definition — Scalar product

Given $\vec u=(u_1,u_2,u_3)$ and $\vec v=(v_1,v_2,v_3)$: $\vec u\cdot\vec v = u_1 v_1 + u_2 v_2 + u_3 v_3.$ The result is a number (a scalar), not a vector.

Property — Geometric meaning

$\vec u\cdot\vec v = |\vec u|\,|\vec v|\cos\theta,$ where $\theta$ is the angle between the two vectors. Consequences:

• Perpendicularity: $\vec u\cdot\vec v = 0 \iff \vec u\perp\vec v$;
• Angle: $\cos\theta = \dfrac{\vec u\cdot\vec v}{|\vec u|\,|\vec v|}$;
• Modulus: $|\vec u|^2 = \vec u\cdot\vec u$.

The scalar product is the tool for computing angles and for recognising perpendicularity: two vectors are orthogonal exactly when their scalar product is zero.

Links

Topics: Analytic geometry in space
Concepts: Scalar product · Vector
Skills: Analytic geometry · Using formulae