Geometric Proof with Vectors (DP IB Analysis & Approaches (AA)) : Revision Note

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Geometric Proof with Vectors

How can vectors be used to prove geometrical properties?

  • If two vectors can be shown to be parallel then this can be used to prove parallel lines

    • If two vectors are scalar multiples of each other then they are parallel

    • To prove that two vectors are parallel simply show that one is a scalar multiple of the other

  • If two vectors can be shown to be perpendicular then this can be used to prove perpendicular lines

    • If the scalar product is zero then the two vectors are perpendicular

  • If two vectors can be shown to have equal magnitude then this can be used to prove two lines are the same length 

  • To prove a 2D shape is a parallelogram vectors can be used to

    • Show that there are two pairs of parallel sides

    • Show that the opposite sides are of equal length

      • The vectors opposite each other will be equal

    • If the angle between two of the vectors is shown to be 90° then the parallelogram is a rectangle

  • To prove a 2D shape is a rhombus vectors can be used to

    • Show that there are two pairs of parallel sides

      • The vectors opposite each other with be equal

    • Show that all four sides are of equal length

    • If the angle between two of the vectors is shown to be 90° then the rhombus is a square

How are vectors used to follow paths through a diagram?

  • In a geometric diagram the vector AB with rightwards arrow on top forms a path from the point A to the point B

    • This is specific to the path AB

    • If the vector AB with rightwards arrow on top is labelled a then any other vector with the same magnitude and direction as a could also be labelled a

  • The vector BA with rightwards arrow on top would be labelled -a

    • It is parallel to a but pointing in the opposite direction

  • If the point M is exactly halfway between A and B it is called the midpoint of A and the vector AM with rightwards arrow on top could be labelled 1 half bold a

  • If there is a point X on the line AB such that AX with rightwards arrow on top equals blank 2 XB with rightwards arrow on top then X is two-thirds of the way along the line AB with rightwards arrow on top

    • Other ratios can be found in similar ways

    • A diagram often helps to visualise this

  • If a point X divides a line segment AB into the ratio p : q then

    •  AX with rightwards arrow on top equals fraction numerator straight p over denominator straight p blank plus blank straight q end fraction AB with rightwards arrow on top

    •  XB with rightwards arrow on top equals fraction numerator straight q over denominator straight p blank plus blank straight q end fraction AB with rightwards arrow on top

How can vectors be used to find the midpoint of two vectors?

  • If the point A has position vector a and the point B has position vector b then the position vector of the midpoint of AB with rightwards arrow on top is 1 half left parenthesis bold a plus bold b right parenthesis

    • The displacement vector AB with rightwards arrow on top equals bold b minus bold a blank

    • Let M be the midpoint of AB with rightwards arrow on top then AM with rightwards arrow on top equals 1 half open parentheses AB with rightwards arrow on top close parentheses equals 1 half open parentheses bold b minus bold a close parentheses

    • The position vector OM with rightwards arrow on top equals blank OA with rightwards arrow on top plus blank AM with rightwards arrow on top equals bold a plus blank 1 half open parentheses bold b minus bold a close parentheses equals 1 half bold b plus 1 half bold a equals 1 half left parenthesis bold a plus bold b right parenthesis

How can vectors be used to prove that three points are collinear?

  • Three points are collinear if they all lie on the same line

    • The vectors between the three points will be scalar multiples of each other

  • The points A, B and C are collinear if AB with rightwards arrow on top equals k BC with rightwards arrow on top

  • If the points A, B and M are collinear and AM with rightwards arrow on top equals blank MB with rightwards arrow on top then M is the midpoint of AB with rightwards arrow on top

Examiner Tips and Tricks

  • Think of vectors like a journey from one place to another

    • You may have to take a detour e.g. A to B might be A to O then O to B

  • Diagrams can help, if there isn’t one, draw one

    • If a diagram has been given begin by labelling all known quantities and vectors

Worked Example

Use vectors to prove that the points A, B, C and D with position vectors a = (3i – 5j – 4k), b = (8i - 7j - 5k), c = (3i - 2j + 4k) and d = (5k – 2i) are the vertices of a parallelogram.

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Amber

Author: Amber

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Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

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