AHL 3.12 (HL)—Vectors

Vectors encode direction and magnitude together.

A vector separates how far something moves from where it points, allowing displacement, velocity and geometric relationships to be combined component by component.

Example

The displacement from A(1,2,0) to B(4,0,6) is (3,−2,6), with length √49=7.

Subtract initial coordinates from final coordinates, then use components for addition and the norm for magnitude.

A position vector is not the same object as a displacement; keep the origin and direction convention explicit.

Component toolkit: for $\mathbf v=(v_1,v_2,v_3)=v_1\mathbf i+v_2\mathbf j+v_3\mathbf k$, $|\mathbf v|=\sqrt{v_1^2+v_2^2+v_3^2}$ and the unit vector in its direction is $\mathbf v/|\mathbf v|$ when $\mathbf v\ne\mathbf0$. If $\overrightarrow{OA}=\mathbf a$ and $\overrightarrow{OB}=\mathbf b$, then $\overrightarrow{AB}=\mathbf b-\mathbf a$; vector equalities can prove midpoint or parallelogram properties component by component.