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Vectors

Vectors

Column vectors and magnitude

Column Vectors Paper 1 + 2 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Column vectors
  4. Magnitude
  5. Position vectors and parallel vectors
  6. Ways they test it
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 7.2 and 7.3: column vectors and magnitude
  • 7.2Write a translation as a column vector and use the notation \(\overrightarrow{AB}\) and \(\mathbf{a}\).
  • 7.2Add and subtract vectors and multiply by a number (scalar).
  • 7.3Find the magnitude (length) of a vector.

Before you start: Pythagoras and translations. Next: Vector geometry.

Formula bank

Add / subtractLearn it\(\begin{pmatrix}a\\b\end{pmatrix}\pm\begin{pmatrix}c\\d\end{pmatrix}=\begin{pmatrix}a\pm c\\b\pm d\end{pmatrix}\)top with top, bottom with bottom.Use it when sums of vectors.
Scalar multipleLearn it\(k\begin{pmatrix}a\\b\end{pmatrix}=\begin{pmatrix}ka\\kb\end{pmatrix}\)Use it when 2a, −3b.
MagnitudeLearn it\(\left|\begin{pmatrix}x\\y\end{pmatrix}\right|=\sqrt{x^{2}+y^{2}}\)Pythagoras.Use it when 'length' or '|a|'.
Between two pointsLearn it\(\overrightarrow{AB}=\) B − Aend minus start.Use it when vector from A to B.
ParallelLearn it\(\mathbf{u}=k\mathbf{v}\)one is a multiple of the other.Use it when 'show that … parallel'.

Column vectors

A vector has a size AND a direction. \(\begin{pmatrix}3\\-2\end{pmatrix}\) means 3 right and 2 down. Vectors are printed in bold (\(\mathbf{a}\)) or with an arrow (\(\overrightarrow{AB}\)); write them underlined by hand.

Triangle law: a then b gives a + baba + b
Follow a, then b: the shortcut is a + b.

Adding vectors means following one then the other. With \(\mathbf{a}=\)\(\begin{pmatrix}3\\-2\end{pmatrix}\) and \(\mathbf{b}=\)\(\begin{pmatrix}-1\\4\end{pmatrix}\): \(\mathbf{a}+\mathbf{b}=\)\(\begin{pmatrix}2\\2\end{pmatrix}\). Subtracting adds the reverse: \(\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})\). Multiplying by a number stretches: \(3\mathbf{a}=\)\(\begin{pmatrix}9\\-6\end{pmatrix}\).

General methodHow to work out any combination like 2a − 3b

  1. Multiply each vector by its number, top and bottom.
  2. Add or subtract the top numbers; then the bottom numbers.
  3. Watch the signs: minus a negative is plus.

Example \(\mathbf{a}=\)\(\begin{pmatrix}3\\-2\end{pmatrix}\), \(\mathbf{b}=\)\(\begin{pmatrix}-1\\4\end{pmatrix}\). Find \(2\mathbf{a}-3\mathbf{b}\).

Scale first, then combine.

  1. \(2\mathbf{a}=\)\(\begin{pmatrix}6\\-4\end{pmatrix}\), \(3\mathbf{b}=\)\(\begin{pmatrix}-3\\12\end{pmatrix}\)
  2. \(6-(-3)=9\), \(-4-12=-16\)

Answer \(\begin{pmatrix}9\\-16\end{pmatrix}\)

Try it yourself

1
\(\mathbf{a}-\mathbf{b}\) for the same a and b?
Show answer

Answer 4, −6

2
\(\mathbf{a}+2\mathbf{b}\)?
Show answer

Answer 1, 6

Magnitude

The magnitude \(|\mathbf{a}|\) is the length of the arrow. The vector is the hypotenuse of a right-angled triangle with sides x and y, so \(|\mathbf{a}|=\sqrt{x^{2}+y^{2}}\). Square first: \((-2)^{2}=4\).

Example Find \(\left|\begin{pmatrix}5\\-12\end{pmatrix}\right|\).

Pythagoras.

  1. \(\sqrt{25+144}=\sqrt{169}\)

Answer 13

Try it yourself

1
\(|\mathbf{a}|\) for \(\mathbf{a}=\)\(\begin{pmatrix}3\\-2\end{pmatrix}\) (2 d.p.)?
Show answer

Answer 3.61

2
\(|\mathbf{a}+\mathbf{b}|\) for \(\mathbf{a}+\mathbf{b}=\)\(\begin{pmatrix}2\\2\end{pmatrix}\) (2 d.p.)?
Show answer

Answer 2.83

Position vectors and parallel vectors

The position vector of a point is the vector from the origin O to it: A(2, 5) has position vector \(\begin{pmatrix}2\\5\end{pmatrix}\). The vector from A to B is 'end minus start': \(\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\).

Example A is \((2,5)\) and B is \((6,-1)\). Find \(\overrightarrow{AB}\) and its length.

B − A.

  1. \(\overrightarrow{AB}=\)\(\begin{pmatrix}4\\-6\end{pmatrix}\)
  2. \(|\overrightarrow{AB}|=\sqrt{16+36}=7.21\)

Answer \(\begin{pmatrix}4\\-6\end{pmatrix}\), length 7.21

Two vectors are parallel when one is a number times the other: \(\begin{pmatrix}6\\-4\end{pmatrix}\) \(=2\)\(\begin{pmatrix}3\\-2\end{pmatrix}\), so they are parallel and the first is twice as long.

Try it yourself

1
\(\begin{pmatrix}k\\6\end{pmatrix}\) is parallel to \(\begin{pmatrix}2\\3\end{pmatrix}\). Find k.
Show answer

Answer 4

2
\(2\)\(\begin{pmatrix}x\\3\end{pmatrix}\) \(+\)\(\begin{pmatrix}1\\y\end{pmatrix}\) \(=\)\(\begin{pmatrix}7\\10\end{pmatrix}\). Find x.
Show answer

Answer 3

Ways they test it

Type 1 Combine vectors

"Find \(2\mathbf{p}-\mathbf{q}\) as a column vector." (2 marks)

Spot it column vectors and letters with numbers.

  1. Scale.
  2. Top with top, bottom with bottom.

Type 2 Magnitude

"Find \(|\overrightarrow{AB}|\)." (2 marks)

Spot it modulus signs or 'length'.

  1. \(\sqrt{x^{2}+y^{2}}\).

For example \(\begin{pmatrix}4\\-6\end{pmatrix}\)

  1. \(\sqrt{52}\)

Answer 7.21

Type 3 Unknowns

"Find the values of m and n." (3 marks)

Spot it letters inside column vectors.

  1. Make a top equation and a bottom equation.
  2. Solve each.

Type 4 Between two points

"Write \(\overrightarrow{PQ}\) as a column vector." (1 mark)

Spot it two coordinates.

  1. End − start.

Mistakes to avoid

  • ✗ Writing a vector as a coordinate (4, −6)✓ column vector
  • ✗ \(\overrightarrow{AB}=\) A − B✓ B − A (end − start)
  • ✗ \(|(3,-2)|=\sqrt{9-4}\)✓ \(\sqrt{9+4}\)Squares are never negative.
  • ✗ Drawing a fraction line between the numbers✓ no line in a column vectorIt is not a fraction.

Summary

  • Column vector = across over up; add/subtract top with top, bottom with bottom.
  • Magnitude \(=\sqrt{x^{2}+y^{2}}\).
  • \(\overrightarrow{AB}\) = B − A.
  • Parallel = a multiple.

If you see …

  • |…|length by Pythagoras.
  • unknowns in vectorstwo equations, top and bottom.

Practice

Practice: easy, medium, hard

Easy

1
\(\mathbf{a}+\mathbf{b}\) for \(\begin{pmatrix}3\\-2\end{pmatrix}\) and \(\begin{pmatrix}-1\\4\end{pmatrix}\)?
Show answer

Answer 2, 2

2
\(3\mathbf{a}\) for \(\mathbf{a}=\)\(\begin{pmatrix}3\\-2\end{pmatrix}\)?
Show answer

Answer 9, −6

3
\(\left|\begin{pmatrix}3\\4\end{pmatrix}\right|\)?
Show answer

Answer 5

Medium

1
\(2\mathbf{a}-3\mathbf{b}\) for \(\begin{pmatrix}3\\-2\end{pmatrix}\) and \(\begin{pmatrix}-1\\4\end{pmatrix}\)?
Guide me

What is it asking? a vector

What do we know? \(\begin{pmatrix}6\\-4\end{pmatrix}\) and \(\begin{pmatrix}-3\\12\end{pmatrix}\)

Plan subtract

Show working
  1. \(\begin{pmatrix}9\\-16\end{pmatrix}\)

Answer 9, −16

2
A(2, 5), B(6, −1). Length AB (2 d.p.)?
Guide me

What is it asking? a length

What do we know? \(\overrightarrow{AB}=\)\(\begin{pmatrix}4\\-6\end{pmatrix}\)

Plan \(\sqrt{52}\)

Show working
  1. 7.21

Answer 7.21

3
\(\begin{pmatrix}k\\6\end{pmatrix}\) is parallel to \(\begin{pmatrix}2\\3\end{pmatrix}\). k?
Guide me

What is it asking? a number

What do we know? 6 = 2 × 3, so multiply by 2

Plan k = 2 × 2

Show working
  1. 4

Answer 4

Hard

1
\(2\)\(\begin{pmatrix}x\\3\end{pmatrix}\) \(+\)\(\begin{pmatrix}1\\y\end{pmatrix}\) \(=\)\(\begin{pmatrix}7\\10\end{pmatrix}\). Find y.
Guide me

What is it asking? a number

What do we know? bottom row: 6 + y = 10

Plan y = 4

Show working
  1. x = 3, y = 4

Answer 4

PartAnswer / workingMarksGuidance
2x + 1 = 7M1
6 + y = 10M1
x = 3, y = 4A1
Total: 3 marks
2
\(\overrightarrow{OP}=\)\(\begin{pmatrix}3\\4\end{pmatrix}\), \(\overrightarrow{OQ}=\)\(\begin{pmatrix}5\\1\end{pmatrix}\). Find \(\overrightarrow{PQ}\).
Guide me

What is it asking? a vector

What do we know? end − start

Plan Q − P

Show working
  1. \(\begin{pmatrix}2\\-3\end{pmatrix}\)

Answer 2, −3

PartAnswer / workingMarksGuidance
\(\begin{pmatrix}2\\-3\end{pmatrix}\)B1
Total: 1 mark
3
\(\left|\begin{pmatrix}8\\-6\end{pmatrix}\right|\)?
Guide me

What is it asking? a length

What do we know? 64 + 36

Plan \(\sqrt{100}\)

Show working
  1. 10

Answer 10

PartAnswer / workingMarksGuidance
10B1
Total: 1 mark

Can I do this?

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

Quiz

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