Vectors in A Level Math Paper 3: common mistakes and examiner advice

Vectors questions in Paper 3 are about lines and the scalar product: writing the equation of a line, deciding whether two lines meet, finding angles and perpendicular distances. The reports show that a vector error in the first line often decides the rest of the question. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Direction errors. Students found BA in place of AB, used OA − OB for AB (the correct form is OB − OA), or used the wrong pair of vectors in a scalar product. Minus signs were frequently lost in the middle of working, and many errors came from copying vectors.
• Position and direction vectors mixed up. Students used the direction of the line instead of the position vector of a point to evaluate a constant, confused OM (a position vector) with AM, or confused the vector AB with the equation of the line through A and B. It was common for students to find the former but not the latter.
• The form of the equation of a line. Many students overlooked the requirement that the equation should begin r = …, left the left-hand side expressed in words, wrote vector AB or l_AB instead of r, or did not use a correct form at all. The left side should be the vector r.
• Checking intersection. The two most common errors were not verifying that all three component equations are satisfied, and showing that two lines intersect without stating the point of intersection. A related slip: using the i and j components to get two equations for the parameters, finding that they disagree, and then not drawing the correct conclusion.
• Parallel and not parallel. Students said that the direction vectors were not parallel without saying which vectors they meant. A related mistake was to think that the equality of a pair of parallel sides was sufficient to prove a particular shape.
• Perpendicular and distance questions. Many students understood that a scalar product was needed but did not use the correct vectors. Others introduced a second general point on the other line, which did not lead to a solution, or found the length from the origin instead of from point A. Sign errors in the scalar product gave the wrong value of the parameter.
• Lengths and magnitudes. Reports mention errors in squaring components when using magnitudes, and leaving out the actual lengths of parts of the figure when expressing vectors such as OM and MN.
• Arithmetic. Sign and arithmetic slips in the working, including when showing that two lines do not intersect.
• Ratio statements. Misunderstanding what BN = 2NC says about the position of N.
• Vertices of a polygon are named in order. Some students did not realise this and worked with the wrong vertices.
• Not knowing how to start. Several reports mention blank responses and students who were not confident working with vectors.

What examiners advise

• Draw a diagram and label the points. Write down each vector in the form "end minus start".
• Use the correct form for the equation of a line, starting with r =.
• State the conclusion explicitly: all three components are satisfied, so the lines meet at the point …, or the third equation fails, so they do not.
• Say which vectors you are comparing when you discuss whether directions are parallel.
• Use the scalar product correctly: the dot product of two directions is zero for perpendicular lines.

Habits that help

• For any vector between two points, write "B minus A" before you substitute the numbers.
• When you find the parameter, substitute it back into both lines to check that the point is the same.
• Remember that a direction vector can be any multiple of the original. Parallel lines need direction vectors that are multiples of each other, not equal vectors.

Questions students ask

Is AB equal to OA − OB?

No. AB is OB − OA, end minus start. Students found BA in place of AB or used OA − OB. Write "B minus A" before you substitute the numbers.

How should I write the equation of a line?

Starting with r = …, using a position vector of a point plus a parameter times a direction vector. Reports record the left-hand side written in words or as "vector AB" and students losing the mark.

How do I show two lines intersect?

Check that all three component equations are satisfied, and state the point of intersection. If the third equation fails, say so, and the lines do not meet.

How do I show lines are parallel?

Say which direction vectors you are comparing, and show that one is a multiple of the other. Parallel lines need direction vectors that are multiples of each other, not equal vectors.

How do I find a perpendicular distance from a point to a line?

Use the scalar product with the correct vectors: it is zero for perpendicular directions. Students lost marks by introducing a second general point on the other line, which did not lead to a solution.