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Vectors

Vectors

Vector geometry

Simple Diagram Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Routes
  4. Points that divide a line in a ratio
  5. Showing parallel and collinear
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 7.4: vector geometry
  • 7.4Find a vector by a route made of known vectors.
  • 7.4Use midpoints and ratios along a line.
  • 7.4Show that lines are parallel or points are collinear.

Before you start: Column vectors and magnitude and simplifying expressions.

Formula bank

RouteLearn it\(\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}=-\mathbf{a}+\mathbf{b}\)going backwards along a vector = minus it.Use it when any vector you don't know.
MidpointLearn it\(\overrightarrow{OM}=\tfrac12(\mathbf{a}+\mathbf{b})\)M is the midpoint of AB.Use it when 'M is the midpoint'.
RatioLearn itAP : PB = m : n \(\Rightarrow\overrightarrow{AP}=\dfrac{m}{m+n}\overrightarrow{AB}\)Use it when 'P divides AB in the ratio…'.
ParallelLearn it\(\overrightarrow{PQ}=k\,\overrightarrow{RS}\)Use it when 'show that PQ is parallel to RS'.
CollinearLearn it\(\overrightarrow{AC}=k\,\overrightarrow{AB}\) AND a shared point AUse it when 'show that A, B, C lie on a straight line'.

Routes

Triangle OAB: OA = a, OB = b, so AB = b − a; M is the midpoint of ABabb − aOABMP
To get from A to B, go back to O, then out to B.

You only know the vectors along some lines (here \(\mathbf{a}\) and \(\mathbf{b}\)). To find any other vector, travel along known lines from its start to its end. Going WITH an arrow adds it; going AGAINST an arrow subtracts it.

General methodHow to find any vector in a diagram

  1. Find the start and the end of the vector you want.
  2. Choose a route between them along lines you know.
  3. Write each step: with the arrow +, against it −. Parts of a line use the ratio (half of AB = ½(b − a)).
  4. Collect the a's and the b's and simplify.

Example \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\). M is the midpoint of AB. Find \(\overrightarrow{AB}\) and \(\overrightarrow{OM}\).

Route O → A → M.

  1. \(\overrightarrow{AB}=-\mathbf{a}+\mathbf{b}=\mathbf{b}-\mathbf{a}\)
  2. \(\overrightarrow{OM}=\mathbf{a}+\tfrac12(\mathbf{b}-\mathbf{a})\)
  3. \(=\tfrac12\mathbf{a}+\tfrac12\mathbf{b}\)

Answer \(\tfrac12\mathbf{a}+\tfrac12\mathbf{b}\)

Try it yourself

1
Same triangle. Find \(\overrightarrow{BA}\) in terms of a and b.
Show answer

Answer a − b

2
OACB is a parallelogram with \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\). Find \(\overrightarrow{OC}\).
Show answer

Answer a + b

Points that divide a line in a ratio

General methodHow to use a ratio on a line

  1. AP : PB = m : n means P is \(\dfrac{m}{m+n}\) of the way from A to B.
  2. \(\overrightarrow{AP}=\dfrac{m}{m+n}\overrightarrow{AB}\).
  3. \(\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{AP}\); expand and collect.

Example P is on AB with AP : PB = 1 : 3. Find \(\overrightarrow{OP}\).

P is ¼ of the way from A to B.

  1. \(\overrightarrow{AP}=\tfrac14(\mathbf{b}-\mathbf{a})\)
  2. \(\overrightarrow{OP}=\mathbf{a}+\tfrac14\mathbf{b}-\tfrac14\mathbf{a}\)

Answer \(\tfrac34\mathbf{a}+\tfrac14\mathbf{b}\)

Try it yourself

1
P is on AB with AP : PB = 2 : 1. Find \(\overrightarrow{OP}\). (Type e.g. a/3 + 2b/3.)
Show answer

Answer ⅓a + ⅔b

Showing parallel and collinear

Vectors are parallel when one is a number times the other. Points are collinear (on one straight line) when two vectors through them are parallel AND share a point.

Example M and N are the midpoints of OA and OB. Show that MN is parallel to AB.

Find both vectors and compare.

  1. \(\overrightarrow{MN}=-\tfrac12\mathbf{a}+\tfrac12\mathbf{b}=\tfrac12(\mathbf{b}-\mathbf{a})\)
  2. \(\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\)
  3. \(\overrightarrow{MN}=\tfrac12\overrightarrow{AB}\)

Answer MN is parallel to AB and half as long

Example \(\overrightarrow{OC}=3\mathbf{b}-2\mathbf{a}\). Show that A, B and C are collinear.

Compare AC with AB.

  1. \(\overrightarrow{AC}=-\mathbf{a}+3\mathbf{b}-2\mathbf{a}=3\mathbf{b}-3\mathbf{a}=3(\mathbf{b}-\mathbf{a})\)
  2. \(\overrightarrow{AC}=3\overrightarrow{AB}\)

Answer AC and AB are parallel and share A, so A, B, C are collinear

Ways they test it

Type 1 Simple route

"Write \(\overrightarrow{AB}\) in terms of a and b." (1 mark)

Spot it a diagram with labelled arrows.

  1. Route along known lines.
  2. Signs by direction.

Type 2 Midpoint or ratio

"P is on AB such that AP : PB = 1 : 3. Find \(\overrightarrow{OP}\)." (2-3 marks)

Spot it a ratio or 'midpoint'.

  1. Fraction of AB.
  2. Add to OA; simplify.

For example AP : PB = 1 : 3

  1. \(\mathbf{a}+\tfrac14(\mathbf{b}-\mathbf{a})\)

Answer \(\tfrac34\mathbf{a}+\tfrac14\mathbf{b}\)

Type 3 Show parallel

"Show that MN is parallel to AB." (2 marks)

Spot it 'show that … parallel'.

  1. Both vectors.
  2. Write one as k × the other, in words.

Type 4 Show collinear

"Show that A, B and C lie on a straight line." (3 marks)

Spot it 'collinear' or 'straight line'.

  1. Two vectors from a SHARED point.
  2. One is k × the other; say so, and name the shared point.

Mistakes to avoid

  • ✗ \(\overrightarrow{AB}=\mathbf{a}-\mathbf{b}\)✓ \(\mathbf{b}-\mathbf{a}\) (end − start)
  • ✗ AP : PB = 1 : 3 used as ⅓ of AB✓ ¼ of AB (1 part out of 1 + 3)
  • ✗ Collinear proof with no shared point✓ say the vectors share a pointParallel alone is not enough.
  • ✗ Not simplifying✓ collect the a's and b'sThe answer mark needs the simplest form.

Summary

  • Any vector = a route along known lines; against an arrow is minus.
  • AP : PB = m : n: \(\overrightarrow{AP}=\frac{m}{m+n}\overrightarrow{AB}\).
  • Parallel: one vector is k × the other. Collinear: parallel + shared point.

If you see …

  • 'in terms of a and b'route.
  • 'show that … straight line'k × and a shared point.

Practice

Practice: easy, medium, hard

Easy

1
\(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\). \(\overrightarrow{AB}\)?
Show answer

Answer b − a

2
M is the midpoint of AB. \(\overrightarrow{OM}\)?
Show answer

Answer ½a + ½b

3
OACB is a parallelogram. \(\overrightarrow{OC}\)?
Show answer

Answer a + b

Medium

1
AP : PB = 1 : 3. \(\overrightarrow{OP}\)?
Guide me

What is it asking? a vector

What do we know? \(\overrightarrow{AP}=\tfrac14(\mathbf{b}-\mathbf{a})\)

Plan add a

Show working
  1. \(\tfrac34\mathbf{a}+\tfrac14\mathbf{b}\)

Answer ¾a + ¼b

2
N is the midpoint of OB. \(\overrightarrow{AN}\)?
Guide me

What is it asking? a vector

What do we know? A → O → N

Plan \(-\mathbf{a}+\tfrac12\mathbf{b}\)

Show working
  1. \(\tfrac12\mathbf{b}-\mathbf{a}\)

Answer ½b − a

3
Q is on BA with BQ : QA = 2 : 3. \(\overrightarrow{OQ}\)?
Guide me

What is it asking? a vector

What do we know? \(\overrightarrow{BQ}=\tfrac25(\mathbf{a}-\mathbf{b})\)

Plan add b

Show working
  1. \(\tfrac25\mathbf{a}+\tfrac35\mathbf{b}\)

Answer ⅖a + ⅗b

Hard

1
M, N are midpoints of OA, OB. \(\overrightarrow{MN}\) is k times \(\overrightarrow{AB}\). k?
Guide me

What is it asking? a number

What do we know? \(\overrightarrow{MN}=\tfrac12\mathbf{b}-\tfrac12\mathbf{a}\)

Plan \(=\tfrac12(\mathbf{b}-\mathbf{a})\)

Show working
  1. ½

Answer 1/2

PartAnswer / workingMarksGuidance
\(\tfrac12(\mathbf{b}-\mathbf{a})\)M1
parallel, half the lengthA1
Total: 2 marks
2
\(\overrightarrow{OC}=3\mathbf{b}-2\mathbf{a}\). Find \(\overrightarrow{AC}\).
Guide me

What is it asking? a vector

What do we know? A → O → C

Plan \(-\mathbf{a}+3\mathbf{b}-2\mathbf{a}\)

Show working
  1. \(3(\mathbf{b}-\mathbf{a})\)

Answer 3b − 3a

PartAnswer / workingMarksGuidance
\(3\mathbf{b}-3\mathbf{a}\)B1
Total: 1 mark
3
So \(\overrightarrow{AC}=k\overrightarrow{AB}\). k?
Guide me

What is it asking? a number

What do we know? \(3(\mathbf{b}-\mathbf{a})\)

Plan \(\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\)

Show working
  1. 3: parallel with A shared, so collinear

Answer 3

PartAnswer / workingMarksGuidance
AC = 3ABM1
common point A, collinearA1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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