Vectors and Their Applications (HL)

Vectors give mathematics a language for quantities that have both size and direction — displacement, velocity, and force. In IB AI HL, vectors model motion, describe lines in space, and let us compute angles and areas efficiently.

Vectors, magnitude, and operations

A vector in two or three dimensions can be written in component (column) form, for example $\mathbf{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix}$. Its magnitude (length) is $|\mathbf{v}| = \sqrt{3^2 + 4^2} = 5$; in three dimensions $|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}$. Vectors add component-wise and scale by multiplying every component. A unit vector in the direction of $\mathbf{v}$ is $\hat{\mathbf{v}} = \frac{1}{|\mathbf{v}|}\mathbf{v}$.

The scalar (dot) product

The dot product of $\mathbf{a}$ and $\mathbf{b}$ is $$\mathb