AHL 3.10 (HL)—Vectors

A vector has magnitude and direction. Two non-zero vectors are parallel when one is a scalar multiple of the other; the scalar's sign tells whether their directions agree or oppose.

Compare components rather than relying on a sketch. The length is √(a·a), and addition represents successive displacements. In 3D, every component must satisfy the same scalar relationship.

(2,−4,6) is 2(1,−2,3), so the vectors are parallel and point the same way. (−2,4,−6) is a negative multiple and points oppositely.

Equal length does not mean parallel, and a single matching component is not enough. Check all components and exclude the zero vector when using direction tests.

A unit vector in the direction of non-zero $v$ is $\hat v=v/|v|$; rescaling to magnitude $s$ gives $s\hat v$. For example, a particle moving at $7\text{ m s}^{-1}$ in direction $3i+4j$ has unit direction $(3i+4j)/5$ and velocity $(21i+28j)/5\text{ m s}^{-1}$. A resultant is the vector sum, and the zero vector has magnitude zero but no defined direction.