Quick questions on Vectors in three dimensions, dot and cross products (QCE Specialist Mathematics Unit 3)

The dot product produces a scalar:

The cross product produces a vector perpendicular to both inputs:

A line through point $A$ with position vector $\mathbf{a}$ and direction $\mathbf{d}$ has vector equation

The dot product is commutative, $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$, and distributes over addition, $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$. A vector dotted with itself returns the square of its length, $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$, which is the algebraic identity behind every magnitude calculation. The base vectors satisfy $\mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} = \mathbf{k} \cdot \mathbf{k} = 1$ and $\mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} = \mathbf{k} \cdot \mathbf{i} = 0$, because they are mutually perpendicular unit vectors. These identities let you expand a product such as $(\mathbf{a} + 2\mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = |\mathbf{a}|^2 + \mathbf{a} \cdot \mathbf{b} - 2|\mathbf{b}|^2$ without resorting to components, a move QCAA rewards in proof-style items.

The cross product is anti-commutative, $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$, so the order matters and reversing it flips the direction of the normal. A vector crossed with itself or with a parallel vector gives the zero vector, $\mathbf{a} \times \mathbf{a} = \mathbf{0}$, which is the algebraic test for parallel vectors. For the base vectors the cyclic rule holds: $\mathbf{i} \times \mathbf{j} = \mathbf{k}$, $\mathbf{j} \times \mathbf{k} = \mathbf{i}$, $\mathbf{k} \times \mathbf{i} = \mathbf{j}$, with a sign change if you reverse any pair. Because $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta$ equals the parallelogram area, half of it is the area of the triangle spanned by the two vectors, which is the standard route to a triangle area from three position vectors.

The single most important decision in a vectors question is which product to use. Reach for the dot product whenever the question mentions an angle, perpendicularity, work done, or a projection, because the dot product converts directly into $\cos\theta$. Reach for the cross product whenever the question mentions a normal direction, an area, or a vector perpendicular to two others, because the cross product converts into $\sin\theta$ and produces a vector. Stating which product you will use and why is itself worth communication credit in extended-response items.