Use 3D vectors with the dot product and cross product to find lengths, angles, projections and areas

What this dot point is asking

SCSA wants confident vector algebra in three dimensions: adding and scaling vectors, computing magnitude and unit vectors, using the scalar product to find angles and scalar/vector projections, and using the vector product to find perpendiculars and the area of a triangle or parallelogram.

Components, magnitude and unit vectors

Write $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$. Then $|\mathbf{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$. A unit vector in the direction of $\mathbf{a}$ is $\hat{\mathbf{a}} = \tfrac{1}{|\mathbf{a}|}\mathbf{a}$. The vector from point $A$ to point $B$ is $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ (position vectors).

The scalar (dot) product

This gives the angle between vectors via $\cos\theta = \tfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$. Two non-zero vectors are perpendicular exactly when $\mathbf{a}\cdot\mathbf{b} = 0$.

The scalar projection of $\mathbf{a}$ onto $\mathbf{b}$ is $\tfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}$, and the vector projection is $\tfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^2}\,\mathbf{b}$.

The vector (cross) product

The result is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ (right-hand rule), with $|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$. It is anticommutative: $\mathbf{a}\times\mathbf{b} = -(\mathbf{b}\times\mathbf{a})$, and $\mathbf{a}\times\mathbf{a} = \mathbf{0}$.