A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 11: Vectors · Lesson 2

Magnitude, direction and position vectors

70 marks

Hints online, answers free with an account:
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Date

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Quick questions to start.

A1
Needed today
Simplify $72 + 50$ .
[2 marks]
Answer
A2
Last lesson
$a = 3 i - 2 j$ and $b = - i + 4 j$ . Find $2 a - 3 b$ .
[2 marks]
Answer
A3
From chapter 9
Find the area of a triangle with sides $5$ cm and $8$ cm and $3 0^{\circ}$ between them.
[2 marks]
Answer $\dots\dots$ cm²
A4
From chapter 10
Solve $tan θ = - 1$ for $0 \leq θ < 36 0^{\circ}$ .
[2 marks]
Answer $θ = \dots\dots$

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

$a = 20 i - 21 j$ and $b = - 4 i + 9 j$ . Find (a) $∣ a ∣$ (b) a unit vector in the direction of $a$ (c) the exact value of $∣ a + 2 b ∣$ .

Hint

∣ x i + y j ∣ = x^{2} + y^{2}

. A unit vector has magnitude

1

: divide the vector by its own magnitude. Combine first, then find the magnitude.

B1
$p = - 12 i + 35 j$ and $q = 6 i - 13 j$ . Find (a) $∣ p ∣$ (b) a unit vector in the direction of $p$ (c) the exact value of $∣ p + 3 q ∣$ .
[5 marks]
Answer(c) $\dots\dots$

Example 2

For $v = - 3 i + 7 j$ , find, to 1 d.p., the angle between $v$ and (a) $i$ (b) $j$ .

Not drawn accurately

Hint The triangle has sides

3

and

7

. The angle with

i

is measured from the positive

x

-axis: here it is over

9 0^{\circ}

B2
For $w = 4 i - 9 j$ , find, to 1 d.p., the angle between $w$ and (a) $i$ (b) $j$ .
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

$a$ has magnitude $14$ and makes $6 0^{\circ}$ with the positive $y$ -axis, as shown. Find $a$ in exact $i$ , $j$ form.

Not drawn accurately

Hint Up:

14 cos 6 0^{\circ}

. Across:

14 sin 6 0^{\circ}

, to the left, so the

i

part is negative.

B3
$b$ has magnitude $18$ , makes $6 0^{\circ}$ with the positive $x$ -axis and points below the $x$ -axis. Find $b$ in exact $i$ , $j$ form.
[3 marks]
Answer $b = \dots\dots$

Practice

A-Level

C1
The point $A$ lies on the circle with equation $x^{2} + y^{2} = 50$ . Given that $O A = 3 k i - k j$ , find the exact possible values of $k$ , and the possible position vectors of $A$ .
[4 marks]
Answer $k = \dots\dots$ $O A = \dots\dots$

C2
The vector $a = p i + q j$ has magnitude $61$ . The angle between $a$ and $i$ is $θ$ , where $cos θ = \frac{11}{61}$ . Find the possible values of $p$ and $q$ .
[4 marks]
Answer $p = \dots\dots$ $q = \dots\dots$
C3
The points $P$ , $Q$ and $R$ have position vectors $(1 - 3)$ , $(62)$ and $(- 2 4)$ . A fourth point $S$ is chosen so that the four points are the vertices of a parallelogram. Find every possible position vector of $S$ .
[4 marks]
Not drawn accurately
Answer $\dots\dots$ $\dots\dots$ $\dots\dots$
C4
$A B C D E F$ is a regular hexagon with centre $O$ . $O A = a$ and $O B = b$ . Find, in terms of $a$ and $b$ , (i) $O C$ (ii) $O E$ (iii) $A C$ (iv) $C F$ .
[4 marks]
Answer(i) $\dots\dots$ (ii) $\dots\dots$ (iii) $\dots\dots$ (iv) $\dots\dots$

Exam-style questions

Edexcel exam style

D1
The vector $v = (k + 1) i + (k - 7) j$ , where $k$ is a constant.
(a)
Show that $∣ v ∣^{2} = 2 k^{2} - 12 k + 50$ .
[2 marks]
(b)
Hence find the least possible value of $∣ v ∣$ , and the value of $k$ at which it occurs.
[3 marks]
Answer $∣ v ∣ = \dots\dots$ $k = \dots\dots$
Total for D1: 5 marks

D2
Ben writes: "To find a unit vector in the direction of $5 i - 2 j$ , divide by $5 - 2 = 3$ . The unit vector is $\frac{5}{3} i - \frac{2}{3} j$ ."
Ben is wrong.
(a)
Show that Ben's vector is not a unit vector, and find the correct unit vector, with rational denominators.
[3 marks]
Answer $\dots\dots$
(b)
The vector $w$ has magnitude $258$ and points in the opposite direction to $5 i - 2 j$ . Find $w$ in the form $a 2 i + b 2 j$ , where $a$ and $b$ are integers.
[3 marks]
Answer $w = \dots\dots$
Total for D2: 6 marks
D3
In triangle $A B C$ , $A B = 6 i + 3 j$ and $A C = 2 i - 5 j$ .
Not drawn accurately
(a)
Find $B C$ , and find $∣ B C ∣$ in simplified surd form.
[3 marks]
Answer $B C = \dots\dots$ $∣ B C ∣ = \dots\dots$
(b)
Find the size of angle $B A C$ , in degrees, to one decimal place.
[4 marks]
Answer $\dots \dots^{\circ}$
(c)
Hence find the area of triangle $A B C$ , giving your answer to $3$ significant figures.
[2 marks]
Answer
Total for D3: 9 marks
D4
The point $P$ lies on the line with equation $2 y = x + 8$ . The position vector of $P$ is $O P = x i + y j$ , where $O$ is the origin, and $∣ O P ∣ = 213$ .
(a)
Show that $5 x^{2} + 16 x - 144 = 0$ .
[3 marks]
(b)
Hence find the two possible position vectors of $P$ .
[3 marks]
Answer $\dots\dots$ or $\dots\dots$
(c)
For the position of $P$ with both coordinates positive, find the angle between $O P$ and $j$ , to one decimal place.
[2 marks]
Answer $\dots \dots^{\circ}$
Total for D4: 8 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The vector $x i + y j$ has integer $x$ , $y$ and magnitude $25$ . How many such vectors are there? How many make an angle under $4 5^{\circ}$ with $i$ ?
[6 marks]
Hint 1 · What to try
You need every integer pair with $x^{2} + y^{2} = 625$ . Try $x = 0, 1, \dots, 25$ .
Hint 2 · The first line
$625 = 0^{2} + 2 5^{2} = 7^{2} + 2 4^{2} = 1 5^{2} + 2 0^{2}$ . Now allow minus signs and swap $x$ and $y$ .
Hint 3 · The full method
$4 + 8 + 8 = 20$ vectors. Under $4 5^{\circ}$ with $i$ needs $x > 0$ and $∣ y ∣ < x$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the magnitude of a vector and a unit vector in its direction.
I can find a vector's direction, and a vector from its magnitude and direction.
I can use position vectors to find lengths and points.
I can find a missing component from a magnitude.

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