AI generatedGeometry and measuresGCSE Higher / IGCSE Extended

Proving collinearity with a single parameter

Proving collinearity with a single parameter.

  • Find two vectors sharing one point, such as AB and AC.
  • Show one is a single scalar multiple of the other, using the same scalar for every component.
  • State both facts: parallel vectors and a common point. Therefore the three points are collinear.

Example 1

a and are non-parallel. OA = a, OB = and OC = ka + . Find so that A, B and C are collinear.

Subtract OA from OB.Subtract OA from OC.The -component fixes the scalar multiple as 2.Match the a-components.Add 1 to both sides.

Then AC = 2AB. The common starting point A establishes collinearity.

Example 2

a and are non-parallel. OA = a, OB = and OC = ka + . Find so that A, B and C are collinear.

Subtract OA from OB.Subtract OA from OC.The -component fixes the scalar multiple as 3.Match the a-components.Add 1 to both sides.

Then AC = 3AB. The common starting point A establishes collinearity.

Your turn

Question 1

a and are non-parallel. OA = a, OB = and OC = ka + . Find so that A, B and C are collinear.

Check answer
Subtract OA from OB.Subtract OA from OC.The -component fixes the scalar multiple as 4.Match the a-components.Add 1 to both sides.

Then AC = 4AB. The common starting point A establishes collinearity.

Question 2

a and are non-parallel. OA = a, OB = and OC = ka + . Find so that A, B and C are collinear.

Check answer
Subtract OA from OB.Subtract OA from OC.The -component fixes the scalar multiple as 5.Match the a-components.Add 1 to both sides.

Then AC = 5AB. The common starting point A establishes collinearity.

Question 3

a and are non-parallel. OA = a, OB = and OC = ka + . Find so that A, B and C are collinear.

Check answer
Subtract OA from OB.Subtract OA from OC.The -component fixes the scalar multiple as 6.Match the a-components.Add 1 to both sides.

Then AC = 6AB. The common starting point A establishes collinearity.

Question 4

a and are non-parallel. OA = a, OB = and OC = ka + . Find so that A, B and C are collinear.

Check answer
Subtract OA from OB.Subtract OA from OC.The -component fixes the scalar multiple as 7.Match the a-components.Add 1 to both sides.

Then AC = 7AB. The common starting point A establishes collinearity.

Question 5

a and are non-parallel. OA = a, OB = and OC = ka + . Find so that A, B and C are collinear.

Check answer
Subtract OA from OB.Subtract OA from OC.The -component fixes the scalar multiple as 8.Match the a-components.Add 1 to both sides.

Then AC = 8AB. The common starting point A establishes collinearity.

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