@ McNotch 2023
Always Learning
VECTOR PROPERTIES OF PLANE SHAPES
Vectors can be used to establish relationships between the line segments in geometric shapes.
We can use these relationships to prove geometrical facts.
1. Triangle Law of Vector Addition.
The length of the lines is found by adding distance around the line starting from initial point.
⃗⃗⃗⃗⃗ of a triangle, then 𝑎 + 𝑏 is
If two vectors 𝑎 and 𝑏 are represented by the sides ⃗⃗⃗⃗⃗
𝐴𝐵 and 𝐵𝐶
⃗⃗⃗⃗⃗ .
represented by the third side. 𝐴𝐶
⃗⃗⃗⃗⃗ = 𝑏, find vector ⃗⃗⃗⃗⃗
Vector ⃗⃗⃗⃗⃗
𝐴𝐵 = 𝑎 and 𝐵𝐶
𝐴𝐶 in terms of 𝑎 and 𝑏.
⃗⃗⃗⃗⃗ = 𝐴𝐵
⃗⃗⃗⃗⃗ + 𝐵𝐶
⃗⃗⃗⃗⃗
𝐴𝐶
=𝑎+𝑏
1|Page
@ McNotch 2023
Always Learning
2. Parallelogram law of addition
⃗⃗⃗⃗⃗ and 𝐴𝐷
⃗⃗⃗⃗⃗ of a
If two vectors 𝑎 and 𝑏 are represented by the adjacent sides 𝐴𝐵
parallelogram, then 𝑎 + 𝑏 is represented by the diagonal of the parallelogram ⃗⃗⃗⃗⃗
𝐴𝐶 .
D
𝑏
C
𝑎+𝑏
A
B
𝑎
⃗⃗⃗⃗⃗
𝐴𝐵 + ⃗⃗⃗⃗⃗
𝐴𝐷 = ⃗⃗⃗⃗⃗
𝐴𝐶 or
⃗⃗⃗⃗⃗
𝐴𝐶 = 𝑎 + 𝑏
EXAMPLE 1
⃗⃗⃗⃗⃗ = 𝑞, M is the midpoint of AB and N divides BC in the ratio
In the diagram, ⃗⃗⃗⃗⃗
𝐴𝐵 = 𝑝 and 𝐵𝐶
1 ∶ 2.
Find in terms of 𝑝 and 𝑞, vector expressions for:
(a)
(d)
2|Page
⃗⃗⃗⃗⃗⃗
𝐴𝑀
⃗⃗⃗⃗⃗⃗
𝑀𝐶
(b)
(e)
⃗⃗⃗⃗⃗⃗
𝐵𝑁
⃗⃗⃗⃗⃗⃗⃗
𝑀𝑁
(c) ⃗⃗⃗⃗⃗⃗
𝐴𝑁
@ McNotch 2023
Always Learning
SOLUTIONS
1
(a) ⃗⃗⃗⃗⃗⃗
𝐴𝑀 = 2 ⃗⃗⃗⃗⃗
𝐴𝐵
1
= 2𝑝
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗ = 1 𝐵𝐶
(b) 𝐵𝑁
3
1
= 2𝑞
(c) ⃗⃗⃗⃗⃗⃗
𝐴𝑁 = ⃗⃗⃗⃗⃗
𝐴𝐵 + ⃗⃗⃗⃗⃗⃗
𝐵𝑁
1
= 𝑝 + 3𝑞
⃗⃗⃗⃗⃗⃗ = 𝑀𝐵
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗ + 𝐵𝐶
(d) 𝑀𝐶
1
= 2𝑝 + 𝑞
⃗⃗⃗⃗⃗⃗⃗ = 𝑀𝐵
⃗⃗⃗⃗⃗⃗ + 𝐵𝑁
⃗⃗⃗⃗⃗⃗
(e) 𝑀𝑁
1
1
= 2𝑃 + 3𝑞
EXAMPLE 2
⃗ 𝐶 = 𝑞, M is the midpoint of AC and the ratio 𝐵𝑁: 𝐵𝐶 is 2: 3.
In the diagram, ⃗⃗⃗⃗⃗
𝐴𝐵 = 𝑝 , 𝐵
Express the following in terms of 𝑝 and/or 𝑞.
(a)
(d)
3|Page
⃗⃗⃗⃗⃗
𝐴𝐶
⃗⃗⃗⃗⃗⃗
𝐴𝑀
(b)
(e)
⃗⃗⃗⃗⃗⃗
𝐵𝑀
⃗⃗⃗⃗⃗
𝐶𝑁
(c) ⃗⃗⃗⃗⃗⃗
𝐴𝑁
@ McNotch 2023
Always Learning
SOLUTIONS
⃗⃗⃗⃗⃗ = 𝐴𝐵
⃗⃗⃗⃗⃗ + 𝐵𝐶
⃗⃗⃗⃗⃗
(a) 𝐴𝐶
=𝑝+𝑞
⃗⃗⃗⃗⃗⃗ = 1 𝐴𝐶
⃗⃗⃗⃗⃗
(b) 𝐴𝑀
2
1
= 2 (𝑝 + 𝑞)
1
1
= 2𝑝 + 2𝑞
(c) ⃗⃗⃗⃗⃗⃗
𝐵𝑀 = ⃗⃗⃗⃗⃗
𝐵𝐴 + ⃗⃗⃗⃗⃗⃗
𝐴𝑀
1
1
= −𝑝 + 2 𝑝 + 2 𝑞
1
1
= 2𝑞 − 2𝑝
⃗⃗⃗⃗⃗ = 3 𝐶𝐵
⃗⃗⃗⃗⃗
(d) 𝐶𝑁
5
3
= 5 (−𝑞)
3
= −5𝑞
(e) ⃗⃗⃗⃗⃗⃗
𝐴𝑁 = ⃗⃗⃗⃗⃗
𝐴𝐶 + ⃗⃗⃗⃗⃗
𝐶𝑁
3
= 𝑝 + 𝑞 − 5𝑞
2
= 𝑝 + 5𝑞
4|Page
@ McNotch 2023
Always Learning
EXAMPLE 3
In the diagram, ⃗⃗⃗⃗⃗
𝐴𝐵 = 𝑝, ⃗⃗⃗⃗⃗
𝐴𝐶 = 𝑞 and M is the midpoint of BC.
Express the following in terms of 𝑝 and/or 𝑞.
(i)
(ii)
(iii)
⃗⃗⃗⃗⃗
𝐵𝐶
⃗⃗⃗⃗⃗⃗
𝐶𝑀
⃗⃗⃗⃗⃗⃗
𝐴𝑀
SOLUTIONS
(i)
Remember ⃗⃗⃗⃗⃗
𝐵𝐴 is a negative vector of ⃗⃗⃗⃗⃗
𝐴𝐵
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
𝐵𝐶
𝐵𝐴 + ⃗⃗⃗⃗⃗
𝐴𝐶
= −𝑝 + 𝑞
=𝑞−𝑝
(ii)
⃗⃗⃗⃗⃗ is a negative vector of 𝐵𝐶
⃗⃗⃗⃗⃗ . Therefore 𝐶𝐵
⃗⃗⃗⃗⃗ = −(𝑞 − 𝑝)
𝐶𝐵
⃗⃗⃗⃗⃗⃗
𝐶𝑀 =
1
⃗⃗⃗⃗⃗
𝐶𝐵
2
1
= 2 (𝑝 − 𝑞)
1
1
= 2𝑝 − 2𝑞
(iii)
⃗⃗⃗⃗⃗⃗
𝐴𝑀 = ⃗⃗⃗⃗⃗
𝐴𝐶 + ⃗⃗⃗⃗⃗⃗
𝐶𝑀
1
1
= 𝑞 + 2𝑝 − 2𝑞
1
1
= 2𝑝 + 2𝑞
5|Page
@ McNotch 2023
Always Learning
EXAMPLE 4
⃗⃗⃗⃗⃗ = 𝑏, the point C is such that 𝐴𝐶 = 3𝐶𝐵 and the point D
In the diagram below, ⃗⃗⃗⃗⃗
𝑂𝐴 = 𝑎, 𝑂𝐵
is such that 𝑂𝐷 = 𝐷𝐴.
(a) Express, in terms of 𝑎 and/or 𝑏 the vectors
⃗⃗⃗⃗⃗ ,
(i)
𝐴𝐵
⃗⃗⃗⃗⃗⃗
(ii)
𝑂𝐷
⃗⃗⃗⃗⃗
(iii)
𝐴𝐶
⃗⃗⃗⃗⃗
(iv)
𝑂𝐶
⃗⃗⃗⃗⃗⃗ , express 𝐵𝑋 in terms of 𝑎, 𝑏 and 𝑘, hence show that
(b) OC and BD meet at ⃗⃗⃗⃗⃗
𝐵𝑋 = 𝑘𝐵𝐷
⃗⃗⃗⃗⃗ = 1 𝑘𝑎 + (1 − 𝑘)𝑏.
𝑂𝑋
2
⃗⃗⃗⃗⃗ , express 𝑂𝑋 in terms of 𝑎, 𝑏 and ℎ.
(c) Given that ⃗⃗⃗⃗⃗
𝑂𝑋 = ℎ𝑂𝐶
⃗⃗⃗⃗⃗ , find the values of ℎ and 𝑘.
(d) Using these two expressions for 𝑂𝑋
SOLUTIONS
(a)
(i)
(ii)
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
𝐴𝐵 = ⃗⃗⃗⃗⃗
𝐴𝑂 + 𝑂𝐵
⃗⃗⃗⃗⃗ = −𝑎 because it is a negative vector of 𝑂𝐴
⃗⃗⃗⃗⃗
Vector 𝐴𝑂
⃗⃗⃗⃗⃗
𝐴𝐵 = −𝑎 + 𝑏
=𝑏−𝑎
1
⃗⃗⃗⃗⃗⃗
𝑂𝐷 = ⃗⃗⃗⃗⃗
𝑂𝐴 because point D is the midpoint of OA.
2
1
= 2𝑎
(iii)
⃗⃗⃗⃗⃗ = 3 𝐴𝐵
⃗⃗⃗⃗⃗ where the ratio of 𝐴𝐶: 𝐶𝐵 is 3: 1
𝐴𝐶
4
3
= 4 (𝑏 − 𝑎)
3
3
= 4𝑏 − 4𝑎
(iv)
6|Page
⃗⃗⃗⃗⃗ = 𝑂𝐴
⃗⃗⃗⃗⃗ + 𝐴𝐶
⃗⃗⃗⃗⃗
𝑂𝐶
@ McNotch 2023
Always Learning
3
3
4
3
4
3
= 𝑎 + 𝑏 − 𝑎 group like terms and simplify
= 𝑎 − 4𝑎 + 4𝑏
1
3
= 4𝑎 +4𝑏
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗
𝐵𝑋 = 𝑘𝐵𝐷
⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗
𝐵𝐷 = 𝐵𝑂
𝑂𝐷
(b)
1
= 2𝑎 −𝑏
1
⃗⃗⃗⃗⃗ = 𝑘( 𝑎 − 𝑏)
Therefore 𝐵𝑋
2
1
= 2 𝑘𝑎 − 𝑘𝑏 where 𝑘 is a constant
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ + ⃗⃗⃗⃗⃗
𝑂𝑋 = 𝑂𝐵
𝐵𝑋 in the direction of BX which involves 𝑘.
1
= 𝑏 + 2 𝑘𝑎 − 𝑘𝑏
1
= 2 𝑘𝑎 + 𝑏 − 𝑘𝑏
1
= 2 𝑘𝑎 + (1 − 𝑘)𝑏 hence shown.
⃗⃗⃗⃗⃗ = ℎ𝑂𝐶
⃗⃗⃗⃗⃗ substitute magnitude of OC from (𝑎)(𝑖𝑣)
(c) 𝑂𝑋
1
3
= ℎ( 𝑎 + 𝑏)
4
4
1
3
= ℎ𝑎 + ℎ𝑏 where ℎ is a constant.
4
4
(d) Formulate simultaneous equations from expression ⃗⃗⃗⃗⃗
𝑂𝑋 = ⃗⃗⃗⃗⃗
𝑂𝑋
1
1
3
𝑘𝑎 + (1 − 𝑘)𝑏 = 4 ℎ𝑎 + 4 ℎ𝑏
2
1
2
1
1
𝑘𝑎 = 4 ℎ𝑎 eliminate magnitude 𝑎 through identity method
1
𝑘 = 4 ℎ … (1) multiply by 4 through out to make it a linear equation.
2
3
((1 − 𝑘)𝑏 = 4 ℎ𝑏 eliminate magnitude 𝑏 through identity method
3
1 − 𝑘 = 4 ℎ --(2) multiply by 4 through out to make it a linear equation.
Therefore, as linear simultaneous equations are:
2𝑘 = ℎ
--(1)
4 − 4𝑘 = 3ℎ
--(1)
2
4
Solve, 𝑘 = 5 and ℎ = 5
7|Page
@ McNotch 2023
Always Learning
EXAMPLE 5
The diagram shows a parallelogram ABCD.
⃗⃗⃗⃗⃗ = 𝑎 and 𝐴𝐷
⃗⃗⃗⃗⃗ = 𝑏. M is the midpoint of CD and N is the point on BC such that 𝐶𝑁 =
Let 𝐴𝐵
𝐵𝐶.
Express vectors and ⃗⃗⃗⃗⃗⃗
𝐶𝑀, ⃗⃗⃗⃗⃗⃗
𝐵𝑁 and ⃗⃗⃗⃗⃗⃗⃗
𝑀𝑁 in terms of 𝑎 and 𝑏.
SOLUTIONS
⃗⃗⃗⃗⃗⃗ as describing a way of getting from C to M moving only along the
1. Think of 𝐶𝑀
directions of 𝑎 and 𝑏.
⃗⃗⃗⃗⃗ = −𝐴𝐵
⃗⃗⃗⃗⃗ .
Going from C to M is the same as going half way from B to A, and 𝐵𝐴
1
1
⃗⃗⃗⃗⃗⃗ = 𝐵𝐴
⃗⃗⃗⃗⃗ = − 𝑎
𝐶𝑀
2
2
2. Going from B to N is twice the distance and in the same direction as from B to C, and
⃗⃗⃗⃗⃗ = 2𝑏.
⃗⃗⃗⃗⃗⃗
𝐵𝑁 = 2𝐵𝐶
3. To get from M to N we go from M to C and then from C to M.
⃗⃗⃗⃗⃗⃗⃗
𝑀𝑁 = ⃗⃗⃗⃗⃗⃗
𝑀𝐶 + ⃗⃗⃗⃗⃗
𝐶𝑁 .
⃗⃗⃗⃗⃗⃗ = −𝐶𝑀
⃗⃗⃗⃗⃗⃗ and 𝐶𝑁
⃗⃗⃗⃗⃗ = 𝐵𝐶
⃗⃗⃗⃗⃗ .
𝑀𝐶
⃗⃗⃗⃗⃗⃗ + 𝐵𝐶
⃗⃗⃗⃗⃗
= −𝐶𝑀
1
= 2𝑎 +𝑏
8|Page
@ McNotch 2023
Always Learning
EXAMPLE 6
The diagram shows a parallelogram ABCD where ⃗⃗⃗⃗⃗
𝐷𝐴 = 𝑎,
(𝑎) Find in terms of 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 the vectors:
⃗⃗⃗⃗⃗
(i)
𝐶𝐵
⃗⃗⃗⃗⃗
(ii)
𝐵𝐶
(iii) ⃗⃗⃗⃗⃗⃗
𝐵𝐷
(b) Find two equivalent expressions for ⃗⃗⃗⃗⃗
𝐴𝐶
SOLUTIONS
(a)
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗
𝐶𝐵
𝐷𝐴
=𝑎
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗
(ii)
𝐵𝐶 = −𝐶𝐵
= −𝑎
⃗⃗⃗⃗⃗⃗
(iii)
𝐵𝐷 = ⃗⃗⃗⃗⃗
𝐵𝐴 + ⃗⃗⃗⃗⃗
𝐴𝐷
= −𝑐 − 𝑎
⃗⃗⃗⃗⃗ = 𝐴𝐵
⃗⃗⃗⃗⃗ + 𝐵𝐶
⃗⃗⃗⃗⃗
(b) 𝐴𝐶
=𝑐−𝑎
(i)
⃗⃗⃗⃗⃗ = 2𝐴𝐸
⃗⃗⃗⃗⃗
Also 𝐴𝐶
= 2𝑏
9|Page
⃗⃗⃗⃗⃗
𝐴𝐸 = 𝑏 and ⃗⃗⃗⃗⃗
𝐴𝐵 = 𝑐.
@ McNotch 2023
Always Learning
Exercise
⃗⃗⃗⃗⃗ = 𝑎 and 𝐴𝐵
⃗⃗⃗⃗⃗ = 𝑏. M is the
1. The diagram shows a parallelogram ABCD with 𝐴𝐵
midpoint of BC and N is the midpoint of CD.
Express the following vectors in terms of 𝑎 and 𝑏.
(i)
(iv)
⃗⃗⃗⃗⃗
𝐵𝐶
⃗⃗⃗⃗⃗⃗
𝑁𝐷
⃗⃗⃗⃗⃗
𝐴𝐶
⃗⃗⃗⃗⃗⃗
𝐴𝑀
(ii)
(v)
⃗⃗⃗⃗⃗
(iii)
𝐶𝐷
⃗⃗⃗⃗⃗⃗⃗
(vi) 𝑀𝑁
2. In the diagram ABCD, ⃗⃗⃗⃗⃗
𝐴𝐵 = 𝑎 and , ⃗⃗⃗⃗⃗
𝐴𝐷 = 𝑏. M is the midpoint of BC, Q is the
1
point on AB such that 𝐵𝑄 = 2 𝐴𝐵 and 𝑃 is the point on the extended line 𝐵𝐶 such that
𝐵𝐶: 𝐶𝑃 = 3: 1, as shown on the diagram below.
(i)
(vi)
10 | P a g e
⃗⃗⃗⃗⃗
𝐴𝑃
⃗⃗⃗⃗⃗⃗
𝑀𝑄
(ii)
(vii)
⃗⃗⃗⃗⃗⃗
𝐴𝑀
⃗⃗⃗⃗⃗⃗
𝐷𝑄
⃗⃗⃗⃗⃗⃗
(iii)
𝑄𝐷
⃗⃗⃗⃗⃗
(vi) 𝑃𝑄
@ McNotch 2023
Always Learning
3.
In the diagram below OABC is a parallelogram, M is the mid-point of ⃗⃗⃗⃗⃗
𝑂𝐴 and ⃗⃗⃗⃗⃗
𝐴𝑋 =
2
⃗⃗⃗⃗⃗ . 𝑂𝐴
⃗⃗⃗⃗⃗ = 𝑎 and 𝑂𝐶
⃗⃗⃗⃗⃗ = 𝑐
𝐴𝐶
7
Express the following in terms of 𝑎 and 𝑐.
⃗⃗⃗⃗⃗⃗
(a) 𝑀𝐴
(b) ⃗⃗⃗⃗⃗
𝐴𝐵
(c) ⃗⃗⃗⃗⃗
𝐴𝐶
⃗⃗⃗⃗⃗
(d) 𝐴𝑋
⃗⃗⃗⃗⃗ = 𝑏 and 𝐵𝐶
⃗⃗⃗⃗⃗ = 𝑎.
4. In the diagram below, 𝑂𝑇: 𝑇𝐶 = 2: 1, ⃗⃗⃗⃗⃗
𝑂𝐴 = 2𝑎, 𝑂𝐵
Express the following vectors in terms of 𝑎 and 𝑏.
⃗⃗⃗⃗⃗
(a) 𝑂𝐶
⃗⃗⃗⃗⃗
(b) 𝑇𝐶
⃗⃗⃗⃗⃗
(c) 𝐵𝑇
⃗⃗⃗⃗⃗
(d) 𝑇𝐴
5. The diagram below is a trapezium OABC. M is the midpoint of AB, OM and CA
⃗⃗⃗⃗⃗ = 4𝑝, 𝑂𝐶
⃗⃗⃗⃗⃗ = 2𝑞 and 𝐶𝐵
⃗⃗⃗⃗⃗ = 2𝑝.
meet at X. 𝑂𝐴
11 | P a g e
@ McNotch 2023
Always Learning
6.
(i)
Express as simply as possible, in terms of 𝑝 and/or 𝑞.
(ii)
⃗⃗⃗⃗⃗
(a) 𝐶𝐴
(b) ⃗⃗⃗⃗⃗
𝐵𝐴
(c) ⃗⃗⃗⃗⃗⃗
𝑂𝑀
⃗⃗⃗⃗⃗ in terms of 𝑝,𝑞 and ℎ.
Given that 𝐶𝑋 = ℎ𝐶𝐴, express 𝐶𝑋
(iii)
Hence, show that ⃗⃗⃗⃗⃗
𝑂𝑋 = 4ℎ𝑝 + 2(1 − ℎ)𝑞
1
⃗⃗⃗⃗⃗⃗ = 𝑞, ⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗ and 𝑃𝑁: 𝑁𝑄 = 3: 1.
In the diagram below, ⃗⃗⃗⃗⃗
𝑂𝑃 = 𝑝 and 𝑂𝑄
𝑂𝑀 = 3 𝑂𝑄
(a) Express in terms of 𝑝 and/or 𝑞.
⃗⃗⃗⃗⃗
(i)
𝑃𝑄
⃗⃗⃗⃗⃗⃗
(ii)
𝑂𝑁
(iii) ⃗⃗⃗⃗⃗⃗
𝑃𝑀
⃗⃗⃗⃗⃗ = ℎ𝑂𝑁
⃗⃗⃗⃗⃗⃗ ,express 𝑂𝑋
⃗⃗⃗⃗⃗ in terms of 𝑝,𝑞 and ℎ.
(b) Given that 𝑂𝑋
1
⃗⃗⃗⃗⃗ = (1 − 𝑘)𝑝 + 𝑘𝑞
⃗⃗⃗⃗⃗ = 𝑘𝑃𝑀
⃗⃗⃗⃗⃗⃗ , show that 𝑂𝑋
(c) Given also that 𝑃𝑋
3
12 | P a g e
@ McNotch 2023
Always Learning
⃗⃗⃗⃗⃗ = 5𝑏 and 𝑅𝑄
⃗⃗⃗⃗⃗ = 3𝑎.
7. In the diagram, ⃗⃗⃗⃗⃗
𝑂𝑃 = 4𝑎, 𝑄𝑅
(a) Express in terms of 𝑎 and/or 𝑏
⃗⃗⃗⃗⃗⃗ ,
(i)
𝑂𝑄
⃗⃗⃗⃗⃗
(ii)
𝑅𝑃
(iii) ⃗⃗⃗⃗⃗
𝑃𝑄
⃗⃗⃗⃗⃗ = 𝑘𝑅𝑃
⃗⃗⃗⃗⃗ , express RX in terms of 𝑎,𝑏 and 𝑘.
(b) OQ and RP meet at X. Given that 𝑅𝑋
Hence show that ⃗⃗⃗⃗⃗
𝑂𝑋 = 5(1 − 𝑘)𝑏 + 4𝑘𝑎
⃗⃗⃗⃗⃗ = ℎ𝑂𝑄
⃗⃗⃗⃗⃗⃗ , express 𝑂𝑋
⃗⃗⃗⃗⃗ in terms of 𝑎,𝑏 and ℎ.
(c) Given also that 𝑂𝑋
(d) Using the two expressions for ⃗⃗⃗⃗⃗
𝑂𝑋, find the values of ℎ and 𝑘.
8. In the diagram, ABO is a triangle in which M is the midpoint of OB and N lies on
1
AB such that 𝐴𝑁 = 5 𝐴𝐵. ON and AM intersect at X.
⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗ = 𝑏.
𝑂𝐴 = 𝑎 and 𝑂𝐵
(a) Express, in terms of 𝑎 and/or 𝑏.
⃗⃗⃗⃗⃗
(i)
𝐴𝐵
13 | P a g e
@ McNotch 2023
Always Learning
⃗⃗⃗⃗⃗⃗
(ii)
𝐴𝑁
⃗⃗⃗⃗⃗⃗
(iii) 𝑂𝑁
⃗⃗⃗⃗⃗⃗
(iv)
𝐴𝑀
1
⃗⃗⃗⃗⃗⃗ , show that ⃗⃗⃗⃗⃗
(b) Given that ⃗⃗⃗⃗⃗
𝐴𝑋 = ℎ𝐴𝑀
𝑂𝑋 = (1 − ℎ)𝑎 + 2 ℎ𝑏
⃗⃗⃗⃗⃗ = 𝑘𝑂𝑁
⃗⃗⃗⃗⃗⃗ , express 𝑂𝑋
⃗⃗⃗⃗⃗ in terms of 𝑎,𝑏 and 𝑘.
(c) If 𝑂𝑋
(d) Use the results of (b) and (c) to find the numerical values of ℎ and 𝑘.
14 | P a g e