0580

Vectors

Vectors and Transformations

Parallel vectors

  • Two vectors are parallel exactly when one is a scalar multiple of the other, so b = ka for some number k
  • To prove it for two expressions, factorise both and show the same bracket falls out of each
    • 6a + 4b factorises to 2(3a + 2b), and 9a + 6b factorises to 3(3a + 2b)
    • So 9a + 6b = 1.5(6a + 4b), which makes them parallel
  • A negative k means parallel but pointing the opposite way
  • Say explicitly that one is a multiple of the other, since that statement is what earns the mark

Collinear points

  • Three points are collinear when they all lie on one straight line
  • Show it by finding two vectors joining the points, showing they are parallel, and noting that they share a common point
  • The shared point is essential: parallel alone only means the lines have the same direction, not that they are the same line
Three points A, B and C shown twice: in the first they lie on one straight line, so AB is parallel to AC and to BC and the points are collinear; in the second they form a triangle, so none of those vectors is parallel and the points are not collinear
Source: Vector problem-solving by Save My Exams
  • The syllabus names showing vectors parallel, showing three points collinear, and ratio and similarity problems as the things to be able to do
A not-to-scale figure with O, A, B and C joined by several lines, where OA is the vector 2a and OC is the vector c, the kind of diagram on which a path such as AB or the ratio in which two lines cut each other has to be worked out
Source: Vector problem-solving by Save My Exams
Exam tip

Write the route you took as a sum of known vectors before simplifying: a correct path earns marks even if the collecting then goes wrong, and any valid route scores, so take the shortest. AB = OB − OA, end minus start — reversing it flips the sign of the whole answer. A ratio m : n divides the line into m + n parts, so the fraction is m ÷ (m + n), never m ÷ n.

Worked example

Showing two vectors are parallel

In a diagram, AB = 2a + 3b and CD = 4a + 6b. Show that AB and CD are parallel.

Solution:

  • Two vectors are parallel exactly when one is a scalar multiple of the other
  • Look for a common factor in CD
  • 4a + 6b = 2(2a + 3b)
  • The bracket is AB, so CD = 2 × AB
  • CD is a scalar multiple of AB, therefore AB and CD are parallel
  • Saying only that "the letters are the same" is not a proof: the statement that must appear is that one vector is a multiple of the other
  • If the two also shared a point, that same working would prove the points collinear — parallel plus a common point is what collinear needs