Vectors — common mistakes

For shortest distance problems, always clearly define your vectors: $\mathbf{p}_1$ and $\mathbf{p}_2$ (points on each line), and $\mathbf{d}_1$ and $\mathbf{d}_2$ (direction vectors). A common error is to mix up these vectors or use an incorrect connecting vector. Drawing a quick sketch can help you visualise the setup and avoid mistakes.

The scalar product, $\mathbf{a} \cdot \mathbf{b}$, results in a scalar (a number). It's calculated as $|\mathbf{a}||\mathbf{b}|\cos\theta$ and is useful for finding angles and projections. The vector product, $\mathbf{a} \times \mathbf{b}$, results in a new vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. Its magnitude is $|\mathbf{a}||\mathbf{b}|\sin\theta$ and is used to find areas and turning effects (moments).

The volume of a pyramid is $\frac{1}{3} \times (\text{base area}) \times (\text{height})$. For a tetrahedron with edges $\mathbf{a}, \mathbf{b}, \mathbf{c}$ from one vertex, the base can be the triangle formed by $\mathbf{b}$ and $\mathbf{c}$, which has area $\frac{1}{2}|\mathbf{b} \times \mathbf{c}|$. The height is the projection of $\mathbf{a}$ onto the normal of this base. Combining these gives $V = \frac{1}{3} \times (\frac{1}{2}|\mathbf{b} \times \mathbf{c}|) \times (\text{height}) = \frac{1}{6}|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$. The total factor is $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$.