How to solve vectors questions in A Level Math Paper 3

Vectors in Paper 3 is a small topic that produces some of the most reliable marks in the paper, and also some of the most feared questions. The questions are built from a short list of ideas: position vectors, the scalar product and the equation of a line. This guide follows the order of the P3 vectors revision workshop on our channel, and shows how those ideas combine and where students most often lose marks.

The ideas every vectors question is built from

• Position vectors. The vector from A to B is the position vector of B minus the position vector of A. Always the second point minus the first. Nearly every question uses this.
• Magnitude is the square root of the sum of the squares of the components. A letter pair with no arrow means a length, so the arrow matters.
• The scalar product multiplies matching components and adds. The only product in the syllabus is the scalar (dot) product. It gives the angle between two vectors, and it is zero exactly when they are perpendicular. A positive value means an acute angle and a negative value an obtuse one.
• Parallel vectors are scalar multiples of each other. To test it, divide matching components, larger over smaller, and check the ratios agree.
• Unit vectors. Divide a vector by its magnitude to get a unit vector in the same direction. For a vector of a given length, scale the unit vector.
• The equation of a line: a point on the line plus a parameter times a direction vector.

Keep to one notation in an answer. Column vectors and i, j, k form should not be mixed within the same vector. In the final answer it is sensible to use the form the question uses.

Finding an angle: check where it is

The scalar product formula gives the angle between two vectors, but which angle you need depends on the question. If the question asks for the angle at a named point, such as angle ABC, the two vectors must either both start at B or both end at B. If one starts and the other ends there, multiply one by −1 first, otherwise you will get the angle on the outside. If the question asks for an acute angle and the calculator gives an obtuse one, subtract it from 180 degrees. In this topic degrees and radians are both acceptable unless the question uses one.

Ratios, midpoints and diagrams

A first part that finds a point from a ratio is often worth only three marks, but the later parts depend on it. Three tools cover it:

• A midpoint: the position vector of the midpoint is half of the sum of the two position vectors.
• Another ratio: when the points lie on one line, the ratio of lengths is also the ratio of the vectors. Write the vector relation, for example AN = 2 NC, replace each vector by a difference of position vectors, and make the unknown position vector the subject. This avoids drawing a diagram.
• Collinear points: if three points lie on a line, the vectors between them are parallel, so one is a multiple of another. This finds unknown constants.

When the question gives a diagram, such as a cube or a roof shape, build the vector as a path from the start to the end along the edges, using the length of each edge times i, j or k. For a parallelogram, label the corners in order around it, then opposite sides are equal as vectors.

Question type 1: Do two lines meet?

Write both lines with different parameters. Equate the three components, which gives three equations in two unknowns. Solve two, then test the values in the third:

• If all three are satisfied, the lines intersect, and you substitute back to get the point.
• If the third fails, and the directions are not parallel, the lines are skew.

The conclusion needs a sentence. Write that the third equation is not satisfied, so there is no intersection. The same idea works backwards: if the question tells you the lines intersect and the lines contain unknown constants, the third equation gives a relation between them. To show that a point lies on a line, put the point equal to the line and check that all three components give the same parameter.

Question type 2: The angle between two lines

Use only the direction vectors, apply the scalar product formula, and take the acute angle if the question asks for the angle between the lines. If the lines are perpendicular, the scalar product of the direction vectors is zero. When a question needs an exact area, find the sine of the angle from its cosine with a right-angled triangle, then use half of ab sin C. A parallelogram is twice a triangle.

Question type 3: Perpendicular from a point to a line

This question is repeated often and is the one most worth memorising. The method is the same each time:

1. Write the position vector of a general point X on the line, in terms of the parameter.
2. Find the vector from the given point P to X.
3. Set its scalar product with the line’s direction vector equal to zero, and solve for the parameter.
4. For the distance, substitute the parameter into PX and find its length. For the foot of the perpendicular, substitute it into the position vector of X.

If the point is the origin, the vector to X is simply the position vector of X. Two common extensions use the foot: a reflection in the line, where the foot is the midpoint of the point and its image, and a point further along the same line from the foot in a given ratio, where you scale the step from P to the foot.

A sign check when two answers appear

If you set an angle equal to a given value and square both sides, you can get two values of the parameter. One gives the required angle and the other gives its supplement. Check the sign of the scalar product: positive means acute and negative means obtuse. Keep the one that matches.

Habits that protect the marks

• Use different letters for the parameters of the two lines. Using the same letter twice is a very common source of lost marks.
• Write the three component equations separately, one per line, and label them.
• Draw a quick sketch of the situation, even a rough one. It shows which angle you are asked for and which two vectors need a scalar product of zero.
• Check a final answer by substituting the point back into the line it should lie on.
• Keep the arithmetic tidy. Vectors questions usually use small whole numbers, so a messy fraction can be a sign of an earlier mistake.