Vectors
Column vectors and magnitude
In this note
- 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
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.
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
- Multiply each vector by its number, top and bottom.
- Add or subtract the top numbers; then the bottom numbers.
- 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.
- \(2\mathbf{a}=\)\(\begin{pmatrix}6\\-4\end{pmatrix}\), \(3\mathbf{b}=\)\(\begin{pmatrix}-3\\12\end{pmatrix}\)
- \(6-(-3)=9\), \(-4-12=-16\)
Answer \(\begin{pmatrix}9\\-16\end{pmatrix}\)
Try it yourself
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Answer 4, −6
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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.
- \(\sqrt{25+144}=\sqrt{169}\)
Answer 13
Try it yourself
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Answer 3.61
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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.
- \(\overrightarrow{AB}=\)\(\begin{pmatrix}4\\-6\end{pmatrix}\)
- \(|\overrightarrow{AB}|=\sqrt{16+36}=7.21\)
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
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Answer 4
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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.
- Scale.
- Top with top, bottom with bottom.
Type 2 Magnitude
"Find \(|\overrightarrow{AB}|\)." (2 marks)
Spot it modulus signs or 'length'.
- \(\sqrt{x^{2}+y^{2}}\).
For example \(\begin{pmatrix}4\\-6\end{pmatrix}\)
- \(\sqrt{52}\)
Answer 7.21
Type 3 Unknowns
"Find the values of m and n." (3 marks)
Spot it letters inside column vectors.
- Make a top equation and a bottom equation.
- Solve each.
Type 4 Between two points
"Write \(\overrightarrow{PQ}\) as a column vector." (1 mark)
Spot it two coordinates.
- 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
Show answer
Answer 2, 2
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Answer 9, −6
Show answer
Answer 5
Medium
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
- \(\begin{pmatrix}9\\-16\end{pmatrix}\)
Answer 9, −16
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
- 7.21
Answer 7.21
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
- 4
Answer 4
Hard
Guide me
What is it asking? a number
What do we know? bottom row: 6 + y = 10
Plan y = 4
Show working
- x = 3, y = 4
Answer 4
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 2x + 1 = 7 | M1 | ||
| 6 + y = 10 | M1 | ||
| x = 3, y = 4 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a vector
What do we know? end − start
Plan Q − P
Show working
- \(\begin{pmatrix}2\\-3\end{pmatrix}\)
Answer 2, −3
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\begin{pmatrix}2\\-3\end{pmatrix}\) | B1 | ||
| Total: 1 mark | |||
Guide me
What is it asking? a length
What do we know? 64 + 36
Plan \(\sqrt{100}\)
Show working
- 10
Answer 10
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 10 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can add, subtract and multiply column vectors.
- I can find the magnitude of a vector.
- I can use position vectors and spot parallel vectors.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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