Vectors
Core concepts
Understand vectors in two and three dimensions as ordered components with magnitude and direction. Be able to add, subtract, and scale vectors, find the scalar (dot) product (\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta), and use it to find angles and test perpendicularity. Know the projection of one vector onto another, the vector equation of a line (\mathbf{r}=\mathbf{a}+t\mathbf{d}), and how to find intersections, angles between lines, and shortest distances. In three dimensions, appreciate that a line and a plane may be parallel, intersecting, or coincident, and handle simple planes via a normal vector.
Traps
Candidates often confuse the dot product (a scalar) with vector multiplication, forget that (\mathbf{a}\cdot\mathbf{b}=0) means perpendicular (not parallel), or lose the parameter range in ray versus line questions. Watch signs in component subtraction, ensure the direction vector is not the zero vector, and remember that the angle between vectors is always between (0) and (\pi). When finding a distance from a point to a line, project the displacement onto the direction and subtract.
Worked example
Find the angle between (\mathbf{a}=(1,2,2)) and (\mathbf{b}=(2,0,1)). Then (\mathbf{a}\cdot\mathbf{b}=2+0+2=4), (|\mathbf{a}|=3), (|\mathbf{b}|=\sqrt5), so (\cos\theta=\frac{4}{3\sqrt5}), giving (\theta\approx53.4^\circ). Rule of thumb: compute the dot product and magnitudes separately before combining.
Exam strategy
Draw or describe the geometry, state the formula used, and keep exact surd forms until the final answer.