UNIT 5 VECTOR

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VECTORS

CONCEPT EXAMPLES

1. Definition

A vector quantity has both magnitude and Velocity, acceleration, displacement

direction

2. Geometrical representation of vectors. aB

Usually represented by a directed line segment. A

AB  a

3. Negative vectors B

A BA   a

4. Zero Vectors AA = 0
Is a vector with zero magnitude and is denoted
by 0 4 Use Phytagoras
3 theorem
5. Magnitude vector
AB =5 unit
Magnitude for vector AB can be written as

AB = a

6. Equal vectors a a = b then
Two vectors are equal if they have the same b
magnitude and direction. a  b and

both vectors
are parallel

EXERCISE 1
1. (a) Find the magnitude for the following vectors.

(b) Draw the negative vectors for each of the following vectors:

1 2
a
c
a

3 4
b
d
a

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2. State a pair of equal vectors from a group of vectors below:
m
ps u v y
ar r x p
p a
p a
a
t
r
pw
aw

p
a

3 ABCDEFGH is a polygon. State the vector a) BA
which is equal to the following vectors:
b) BC
F E c) ED
G d) AH

D

H
C

AB

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CONCEPT EXAMPLE
7. Multiplying vectors by scalars

ka is a vector with a magnitude that
is k times of a.

AB  2 a

PQ  1 a
2

RS  3 a
2

EXERCISE 2

1. Given a vector p in the diagram below. Construct the following vectors:

a) AB = 2 p (b) CD = 3 p (c) EF = 3 p (d) GH =  1 p
2 2

p

2. State the following vectors in terms of vector a:

C E GJ
a B

F H
D

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3. ABCD is a uniform hexagon such that AB = a and BC = b, D
State the following vectors in terms of a and b:
(a) DE C
E A
(b) CF B

(c) EF F
(d) BE A

4. PQR is a straight line such that PQ : PR = 2 : 5 and PQ = u .
State the following vectors in terms of u.

(a) QR

(b) QP

(c) PR

5. (b) Given PQ = 4 RS and
a) Given AB = 3 CD and 7

2 RS  21 cm .

AB  6 cm . Find PQ

Find CD

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CONCEPT EXAMPLE
8. Parallel vectors
ab b =2a ,
a) Given vectors u and v then b is
(i) If u = k v, where k is a non-zero scalar, parallel to a

then u and v are parallel.

(ii) If u and v are parallel, then u = k v,
where k is a non-zero scalar.

EXERCISE 3

1. Given that PQ = 4a + 7 b, find the following vectors:
(a) 7 PQ

(b) 4 PQ
5

(c) 5 PQ

2.

(a) Given that PQ = 3a + m b and (b) Given that AB = 2 a + 5 b and

XY = n a + 6 b, find the value of m CD =  12 a  6 b, find the value of k
and n such that 2PQ = 5 XY 5

such that 3AB = k CD

3. (b) Given that MN = 6 v and
(a) Given that AB = 4 u and
CD =  5 u , determine whether PQ =  2 v , determine whether
2 3
vectors AB and CD are parallel.
vectors MN and PQ are parallel.

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(c) Given that XY = 2a + 3b and (d) Given that AB =  2a + b and
CD = 6a  3 b , determine whether
KL = a + 3 b , determine whether vectors AB and CD are parallel
2

vectors XY and KL are parallel.

4. Given that , 4DE = 3BC , what can you say about the line segments DE and BC? C

a) If DE  6 , find BC . E

A DB

b) Given AE  3 b , state AC in terms of b .

~~

CONCEPT EXAMPLE
8(b) Given that PQ  3u and QR 5u , show
If EF  k FG , k is a non-zero scalar, then that P, Q and R are collinear

(i) EF is parallel to FG Solution:

(ii) E,F, and G are collinear.

PQ  3u. , QR  5u.

 u  1 PQ ,  u  1 QR
35

 1 PQ  1 QR
35

PQ  3 QR
5

 P, Q and R are collinear.

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EXERCISE 4 (b) Given that MN = 4v and NR = 7v ,
determine whether vectors M, N, and R
1. are collinear.
(a) Given that PQ = 3u and QR = 5u ,
determine whether vectors P, Q, and R
are collinear.

CONCEPT EXAMPLE
9. If m a = n b ,  m = n = 0 when Given (m  2) x = (2n +5)y , m and n are
scalars. Find the values of m and n if x and y
(i) vectors a and b are not parallel are not parallel and are not zero vectors.
(ii) vectors a and b are not zero vectors.

Solution: , 2n +5 = 0
m2=0 , n = 5

m=2 2

EXERCISE 5

1. Given that (2h + 3) a = (2k – 5 ) b and a and b are non-parallel vectors, find the numerical
values of h and k .

2. Given that (k + 2) x = (2h – 3) y and x and y are non-parallel vectors, find the numerical
values of h and k .

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CONCEPT EXAMPLES
10. ADDITION AND SUBTRACTION OF
(i) a  2a  3 a
VECTORS (ii) 4a  3a = a
1. Addition and subtraction of two parallel

vectors

2. Addition of two non-parallel vectors: C
a) Triangle law
b) Parallelogram Law a+b

3. Addition of three or more vectors. b AB + BC = AC
a) Polygon Law AaB

4. Subtraction of vectors. D aC
subtraction of two non-parallel vectors. b a+b

b

AB
AB + AD = AC

AB + BC + CD + DE = AE
D

E

C

AB

u
v

u v u
-v

EXERCISE 6

1. Draw the following vectors in the space provided:

(a) a + b (b) a  2 b

a a
b

b

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2. Given AB = a and AC = b. M is a mid point of BC. A C

Find
(a) BC

(b) BM

(c) AM

3. PQRSTU is a regular hexagon with center O. T S
Express each of the following as single vector: T R
(a) PQ + PT U
Q
(b) RS + ST P

(c) PQ + PR + PT + PU

4. ABCD is a parallelogram. Diagonal AC and BD intersect at point O. B
Find the resultant vector of each of the following:
A
(a) AB + BD

(b) CO + OD O
(c) CA + BC CD

(d) OB + DO

5. Express the following vectors in terms of a and b : P
(i) OP R
B
(ii) OR
A

ba
O

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5. Given OX = 6 x , OY = 4 y. Express the following vectors in terms of x and y .

(a) OY  XO Y

4y X
O
6x
(b) XY

6. In the given diagram , PQ = a , PR = b and RS =  2a.  2a. R
Express each of the following in terms of a and b . S
(a) QR
b Q
(b) PS
a
P

7. In the given diagram, T is a midpoint of RQ. Find the following vectors in terms of a and b :

(a) PT R

T

2b

P Q
6a
(b) PR

8. In a diagram , D is a mid point of CE. Given CE = 4AF = 3x and EF = 2CB = 2y.
Express each vector in terms of x and y:

(a) AE A F
C

(b) AD

BD E

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CONCEPT EXAMPLES
11. REPRESENT VECTORS AS A D

COMBINATION OF OTHER VECTORS. E

C

A DA B

AB = AC + CB
AB = AE + ED + DC + CB
AB = AC - BC

EXERCISE 7 Q

1. Given PQ = p and PR = q. M is a mid point R
of QR
PD A
. Find
(a) QR

(b) PM

2. Given AB = x, BC = y and AM = 2 AC . B
3

Find
(a) AC

(b) AM A C
(c) BM M

3. ABC is a triangle. BA = 4u and BC = 3v. C B
P is on the line AC and Q is on the line AB P Q
such that AP = 3 PC and AQ = 3 AB.
25 A
Express each vector in terms of u and v:
(a) CA

(b) AP

(c) PQ

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4. In the diagram , C is a midpoint of AB and
A DC B
D is a midpoint of AC. If OA = a and AA A
OB = b, express each of the following
vectors in terms of a and b. ab

(a) AB O
A
(b) AD

(c) OD

(d) OC

1 a + 1 b , show that ED is parallel to OA .
If OE =
44

5. Given OA = 3p + q , OB = 3p + 4q and OC =  3p + 5q . OA is extended to a point D such that
OD = m OA and CD = n CB. Express OD in terms of

(a) m , p and q

(b) n , p and q

Hence, find the values of m and n.

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6. In the diagram, given A is a point such that 3OA = OC and B is the mid point of OD.

The straight lines AD and BC intersect at point P.

Given that OA = 3a and OB = 5b .

C

(a) Express in terms of a and b

(i) AD 3a A

O P
5b
(ii) BC

B

(b) Given that AP = h AD and BP = k BC. D
Express OP
(i) in terms of h, a and b

(ii) in terms of k, a and b .

Hence, or by using other method, find the values of h and k.

(c) Find BP : PC

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CONCEPT EXAMPLE

12. VECTORS IN CARTESAN PLANE Q PQ = 3 i + 4 j
=  43
1. Vectors in the form x i + y j and  x  4
y P3

2. Magnitude vectors = x 2  y2 PQ  32  42  25  5 unit

3. Unit vectors in given directions Unit vector in the direction of OA if A( 2,3) is

If r = x i + y j , then the unit vector is OA  2 i  3j 2i  3j

r r xi  y j OA 22  32
rˆ = ~   13
~r ~

~ x2  y2 x2  y2

4. Adding or subtracting two or more vectors a = 5 i + 4j =  5 
4
If a = x1 i + y1 j =  x1 
y1
b = 4 i + j =  14
b = x2 i + y2 j =  x 2 
y 2 (i) a + b =  54 +  14 = 15 = i + 5 j

a+b =  x 1    x 2  =  x1  x 2   5   14  174 = 14 i + 7 j
y 1 y 2 y1  y 2 4
(ii) 2a b= 2  =

5. Multiply vectors by scalars (iii) a + 2b =  5  + 2  14
4
If a=xi+yj=  x 
y
= 5 4(28)   63 = 3 i + 6 j
ka = k  x  =  kx  , k is a constant.
y ky

EXERCISE 8

1. Express each of the vectors shown in the diagram in the form of

(a) x i + y j (b)  x  y
y
a c
d x

O

b

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2. Find the magnitude of each of the following vectors:

(a) p = 4 i – 5 j (b) r = –7 i –3 j (c) q =   2 
6

3. Given that a = 3 i – 5 j and b = 2 i + p j . Find the value of p if

(a) a – 2 b = – i – 11 j (b) 2a  b  41

4. Given a = i + 3 j and b = 4i – 2j . Find

(a) a + b (b) a – b (d) 2a – 5b

5. b) Determine the unit vector rˆ
if r = – 2 i +3 j
a) Determine the unit vector rˆ
if r = 6 i – 8 j

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6. Given a point A( 5, 12) and point B(– 4,3) , express vectors OA and OB in terms of i and j .
Find the unit vector in the direction of OA.

7. Given a point P(1,8) and point Q(7,0). Find

(a) vector PQ in the form x i + y j and  x 
y

(b) the magnitude of PQ

(c) the unit vector in the direction of PQ .

8. Given OA = 4 i + 5 j , OB = h i + 5h j and AB = –2 i + k j , find the numerical value of h
and k.

9. Given vectors r = 4 i – 6 j and s = 6 i + k j . If r and s are parallel vectors,
find the value of k.

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10. Given AB = – 6 i + 8 j and CD = h i + 2 j .
Find the value of h such that AB is parallel to CD.

11. Given OA = – 3 i + 2 j , OB = 5 i + 6 j and OC = i + 4 j .
(a) Determine the following vectors , in terms of i and j.
(i) AB

(ii) AC

(b) Show that A , B and C are collinear.

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