Vectors in two dimensions in Additional Math 0606: common mistakes and examiner advice

Vector questions in Additional Math cover position vectors, magnitudes, ratios, velocity and motion, and geometry in terms of a and b. The reports show that the errors are the same in many questions: signs, the form of the answer, and what the question is asking for. 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

• Adding when subtracting is needed. Summing the position vectors of P and Q instead of finding the difference, to get PQ. Some did not know that AB is the negative of BA and tried to "fudge" the result.
• Position vectors with time. Giving a position vector that did not include t or the initial position, such as a vector that did not start from the initial position or offering (18i + 21j) + t(8i + 8j) when it was not what was asked. In another question, students confused the initial position vector with the velocity vector.
• Form of the answer. Writing vectors in an incorrect form, such as a column vector with i and j inside it, or mixing the two.
• Velocity and speed. Forgetting to negate the x-component of the velocity vector, finding the correct velocity vector, doubling it and not adding it to the starting position, converting surds to harshly rounded decimals, and not understanding that speed is the magnitude of the velocity vector.
• Ratios. Misreading a ratio (taking AP : AB as 1 : 3), direction errors, and multiplying by λ only when a vector is a fractional part of another. Others multiplied by 2 + λ or 1 + λ.
• Magnitudes and directions. Adding the given direction vectors and ignoring the given magnitudes, or finding the magnitude of an earlier answer when the vector itself was needed.
• Vectors in a and b. Taking OY = ½(a − b) instead of ½(a + b), errors in the direction of vectors, sign slips when forming ZY, and difficulty combining the vectors into a single equation in a and b and equating coefficients. Reports say many could not use a condition such as YZ = λXY.
• Sign errors that hide. Sign errors in the components can give a magnitude and bearing that look right but come from an incorrect vector, and earn only method marks.
• Given results. Not obtaining the given result should mean that previous work needs to be checked, rather than contriving to obtain it.
• Not finishing. Some candidates found OC and did not complete the solution. Others misread their own writing, such as using OD as AD.
• Over-simplifying. Simplification that is not required often caused errors in a later part.
• Practice. Candidates often did not respond to one or more parts, and the reports say they would benefit from practising vector questions.

What examiners advise

• Write each vector in terms of known ones by following a path, and check the direction of each step.
• Give vectors in a consistent form (i and j, or column), and include t if the question has a time variable.
• Keep the answer unsimplified when simplification is not required.
• Check the result against the question. When a difference in velocity vectors is unexpectedly large, look for an error.
• If you do not obtain the given result, go back and check your earlier work.
• Read the question again at the end to make sure you have answered every part.

Habits that help

• Draw a quick sketch with the points labelled and arrows for each vector.
• Write "final − initial" every time you find a displacement between two points.
• When you equate vectors, write two equations (one for each component or coefficient) and label them.

Questions students ask

How do I find the vector PQ?

Subtract: PQ = OQ − OP, end minus start. Some students summed the position vectors. AB is the negative of BA.

What form should I give vectors in?

A consistent one: column vectors, or i and j. Do not put i and j inside a column vector. Include t if the question has a time variable and start from the initial position.

How do I find a position vector from a ratio?

Read the ratio carefully: taking AP : AB as 1 : 3 is a misreading. Write each vector in terms of known ones by following a path and check the direction of each step.

Why does the question give magnitudes and directions?

Use both. Students added the given direction vectors and ignored the given magnitudes, or found the magnitude of an earlier answer when the vector itself was needed.

What should I do if I cannot reach the given result?

Go back and check your earlier work instead of contriving to obtain it. Sign errors can hide: a magnitude and bearing may look right but come from an incorrect vector and earn only method marks.