Study case: proving two vectors are parallel In this GCSE Vectors Geometric Proof Problem Tutor, I use one focused study case to show how vector proof can be made logical and manageable. The problem is: if vector AB = 2a + b and vector CD = 6a + 3b, prove that AB is parallel to CD. The answer is that CD = 3AB, so the two vectors are scalar multiples and therefore parallel. I like this example because it teaches the key GCSE vector proof idea: if one vector is a multiple of another, the lines are parallel. Before solving, I ask the student to identify what needs to be proved. We are not asked to find a length or an angle. We are asked to prove a relationship between two vectors. That means the final answer must include a reason, not just a calculation. I encourage students to write short, clear proof sentences because vector questions often award marks for reasoning. Method 1 is the scalar multiple method. We compare AB = 2a + b with CD = 6a + 3b. I factor 3 out of CD to get CD = 3(2a + b). But 2a + b is AB, so CD = 3AB. This means CD has the same direction as AB and is three times as long. Therefore AB and CD are parallel. This is the cleanest exam method because it links the algebra directly to the geometric conclusion. Method 2 is the coefficient comparison method. I ask the student to compare the coefficient of a and the coefficient of b. In AB, the coefficients are 2 and 1. In CD, the coefficients are 6 and 3. Both have been multiplied by 3. Because both components have the same scale factor, the vector direction has not changed. This method helps students see why it would not be enough for only one coefficient to be a multiple. Both parts of the vector must scale together. Method 3 is the diagram interpretation method. I explain that if a vector is multiplied by a positive number, it stretches in the same direction. If it is multiplied by a negative number, it points in the opposite direction but is still parallel. In this case, CD = 3AB, so CD is in the same direction as AB. This visual explanation is useful for students who understand the algebra but are unsure how it proves a geometric statement. After the three methods, I compare them with the student. The scalar multiple method is quickest. The coefficient method is a useful check. The diagram method builds understanding. In tutoring, I want the student to connect all three so that a vector proof is not just a formula. It becomes a statement about direction, scale and geometry. The most common mistakes are saying the vectors are equal when they are only parallel, forgetting to factor out the same number from every term, using length language when the question asks for direction, and not writing a final proof sentence. I use these mistakes to guide the lesson. If a student writes CD = AB, we revisit the scale factor. If they stop at CD = 3AB, I ask them to explain what that means geometrically. This topic is important for GCSE and IGCSE because vector proof questions often combine midpoint, ratio, parallel lines and collinearity. The same scalar multiple idea appears in many forms. A student might need to prove that two lines are parallel, that three points are on a straight line, or that a quadrilateral is a trapezium. The basic idea is the same: compare vectors and look for a common scale factor. A useful extension is to prove that points P, Q and R are collinear by showing PQ is a multiple of QR. Another extension is to use position vectors from an origin and form a vector between two points by subtracting position vectors. These extensions help students move beyond a simple proof and prepare for more complex exam questions. In one-to-one tutoring, I would finish by giving the student a similar proof with different vectors. I would ask them to state the scale factor and then write the conclusion in words. I want them to say: because one vector is a scalar multiple of the other, the lines are parallel. That sentence shows the proof is understood. For support with this topic, students can book one-to-one tutor services: https://www.mastermathstutoring.co.uk/services. For the wider course page, visit GCSE and IGCSE Maths tutoring: https://www.mastermathstutoring.co.uk/gcse-igcse. To ask about lessons or availability, use contact MasterMaths Tutoring: https://www.mastermathstutoring.co.uk/contact-me.