almathematicsvectorsadditionalmathe ... is a mid point of bc. find (a) bc (b) bm (c) am 3. pqrstu is...

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VECTORS

Name

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Additional Mathematics Vectors

VECTORS

CONCEPT EXAMPLES

1. Definition

A vector quantity has both magnitude and

direction

Velocity, acceleration, displacement

2. Geometrical representation of vectors.

Usually represented by a directed line segment.

3. Negative vectors

4. Zero Vectors

Is a vector with zero magnitude and is denoted

by 0

5. Magnitude vector

Magnitude for vector AB can be written as

AB = a

6. Equal vectors

Two vectors are equal if they have the same

magnitude and direction.

EXERCISE 1

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

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

1

2

3

4

A

B a

aAB

B

aBA

Use Phytagoras

theorem

AB =5 unit 3

4

a

b

a = b then

ba and

both vectors

are parallel

d

a

A

AA = 0

a c

a

b


2. State a pair of equal vectors from a group of vectors below:

3 ABCDEFGH is a polygon. State the vector

which is equal to the following vectors:

a)

b)

c)

d)

BA

BC

ED

AH

p

a s

r

p

a t

r

p

a

m

y

p

a

u

r

p

a

v

w

w

p

a

x

B A

H C

D

E F

G


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 = 2

3 p (d) GH = 2

1 p

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

CONCEPT EXAMPLE

7. Multiplying vectors by scalars

ka is a vector with a magnitude that

is k times of a.

a2AB

a2

1PQ

a2

3RS

p

a

J

B

C

D

E

F

G

H

a

B

P

A Q

P

A

R

P

A

S

A

A


3. ABCD is a uniform hexagon such that AB = a and BC = b,

State the following vectors in terms of a and b:

(a) DE

(b) CF

(c) EF

(d) BE

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.

a) Given AB = 2

3 CD and

CDFind

.cm6AB

(b) Given PQ =

7

4RS and

PQFind

.cm21RS

A B

C

A

D E

F


EXERCISE 3

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

(a) 7 PQ

(b) 5

4PQ

(c) 5 PQ

2.

(a)

Given that PQ = 3a + m b and

XY = n a + 6 b, find the value of m

and n such that 2PQ = 5 XY

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

CD = 5

12 a 6 b, find the value of k

such that 3AB = k CD

3.

(a)

Given that AB = 4 u and

CD = 2

5 u , determine whether

vectors AB and CD are parallel.

(b) Given that MN = 6 v and

PQ = 3

2 v , determine whether

vectors MN and PQ are parallel.

CONCEPT EXAMPLE

8. Parallel vectors

a) Given vectors u and v

(i) If u = k v, where k is a non-zero scalar,

then u and v are parallel.

(ii) If u and v are parallel, then u = k v,

where k is a non-zero scalar.

a b b = 2 a ,

then b is

parallel to a


(c)

Given that XY = 2a + 3b and

KL = a + 2

3 b , determine whether

vectors XY and KL are parallel.

(d) Given that AB = 2a + b and

CD = 6a 3 b , determine whether

vectors AB and CD are parallel

4. Given that , 4DE = BC3 , what can you say about the line segments DE and BC?

a) If 6DE , find .BC

b) Given ~b3AE , state AC in terms of

~b .

CONCEPT EXAMPLE

8(b)

If FGkEF , k is a non-zero scalar, then

(i) EF is parallel to FG

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

Given that u3PQ and u5QR , show

that P, Q and R are collinear

Solution:

.u3PQ , .u5QR

PQ3

1u , QR

5

1u

PQ3

1QR

5

1

PQ QR5

3

P, Q and R are collinear.

C

E

A D B


EXERCISE 4

1.

(a)

Given that PQ = 3u and QR = 5u ,

determine whether vectors P, Q, and R

are collinear.

(b) Given that MN = 4v and NR = 7v ,

determine whether vectors M, N, and R

are collinear.

CONCEPT EXAMPLE

9. If m a = n b , m = n = 0 when

(i) vectors a and b are not parallel

(ii) vectors a and b are not zero vectors.

Given (m 2) x = (2n +5)y , m and n are

scalars. Find the values of m and n if x and y

are not parallel and are not zero vectors.

Solution:

m 2 = 0 , 2n +5 = 0

m = 2 , n = 2

5

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 .


CONCEPT EXAMPLES

10. ADDITION AND SUBTRACTION OF

VECTORS

1. Addition and subtraction of two parallel

vectors

2. Addition of two non-parallel vectors:

a) Triangle law

b) Parallelogram Law

3. Addition of three or more vectors.

a) Polygon Law

4. Subtraction of vectors.

subtraction of two non-parallel vectors.

(i) a3a2a

(ii) 4a 3a = a

AB + AD = AC

AB + BC + CD + DE = AE

EXERCISE 6

1. Draw the following vectors in the space provided:

(a) a + b (b) a 2 b

A a

b

a + b

b

a + b b

a

A B

C D

B

C

AB + BC = AC

D

A

C

B

E

u v

u

- v u v

a

b

a

b


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

Find

(a) BC

(b) BM

(c) AM

3. PQRSTU is a regular hexagon with center O.

Express each of the following as single vector:

(a) PQ + PT

(b) RS + ST

(c) PQ + PR + PT + PU

4. ABCD is a parallelogram. Diagonal AC and BD intersect at point O.

Find the resultant vector of each of the following:

(a) AB + BD

(b) CO + OD

(c) CA + BC

(d) OB + DO

5. Express the following vectors in terms of a and b :

(i) OP

(ii) OR

a b O

B

R

P

A

B

A

C

R

P Q

S T

T

U

A

C

B

D

O


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

(a) OY XO

(b) XY

6. In the given diagram , PQ = a , PR = b and RS = 2a.

Express each of the following in terms of a and b .

(a) QR

(b) PS

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

(a) PT

(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

(b) AD

6 x

4 y

X O

Y

R

P

S

Q

a

b

2a.

6a P

R

Q

T

2 b

B

C

E

A

D

F


CONCEPT EXAMPLES

11. REPRESENT VECTORS AS A

COMBINATION OF OTHER VECTORS.

AB = AC + CB

AB = AE + ED + DC + CB

AB = AC - BC

EXERCISE 7

1. Given PQ = p and PR = q. M is a mid point

of QR

. Find

(a) QR

(b) PM

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

2AC .

Find

(a) AC

(b) AM

(c) BM

3. ABC is a triangle. BA = 4u and BC = 3v.

P is on the line AC and Q is on the line AB

such that AP = 2

3PC and AQ =

5

3AB.

Express each vector in terms of u and v:

(a) CA

(b) AP

(c) PQ

D A

C

D

A B

E

Q

D A

R

P

B

A C M

B

A

C

P Q


4. In the diagram , C is a midpoint of AB and

D is a midpoint of AC. If OA = a and

OB = b, express each of the following

vectors in terms of a and b.

(a) AB

(b) AD

(c) OD

(d) OC

If OE = 4

1a +

4

1b , show that ED is parallel to OA .

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.

B

A

b

C

A A D

A

O

A

a


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 .

(a) Express in terms of a and b

(i) AD

(ii) BC

(b) Given that AP = h AD and BP = k BC.

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

D

O

A

B

C

P

5b

3a


CONCEPT EXAMPLE

12. VECTORS IN CARTESAN PLANE

1. Vectors in the form x i + y j and

y

x

2. Magnitude vectors = 22 yx

unit52543PQ 22

3. Unit vectors in given directions

If r = x i + y j , then the unit vector is

~r̂ =

~

~

r

r

2222

~

yx

jyix

yx

r

Unit vector in the direction of OA if A( 2,3) is

13

j3i2

32

j3i2

OA

OA

22

4. Adding or subtracting two or more vectors

If a = x1 i + y1 j =

1

1

y

x

b = x2 i + y2 j =

2

2

y

x

a + b =

1

1

y

x

2

2

y

x =

21

21

yy

xx

a = 5 i + 4j =

4

5

b = 4 i + j =

1

4

(i) a + b =

4

5+

1

4 =

5

1= i + 5 j

(ii) 2a b = 2

4

5

1

4=

7

14= 14 i + 7 j

5. Multiply vectors by scalars

If a = x i + y j =

y

x

k a = k

y

x =

ky

kx , k is a constant.

(iii) a + 2b =

4

5+ 2

1

4

=

6

3

24

)8(5 = 3 i + 6 j

EXERCISE 8

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

(a) x i + y j (b)

y

x

a

b

c

O x

y

3

4

P

PQ = 3 i + 4 j

=

4

3

Q

d


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

(a) p = 4 i – 5 j (b) r = –7 i –3 j (c) q =

6

2

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) 41ba2

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

(a) a + b

(b) a – b

(d) 2a – 5b

5.

a) Determine the unit vector r̂

if r = 6 i – 8 j b) Determine the unit vector r̂

if r = – 2 i +3 j


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

y

x

(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.


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.

almathematicsvectorsadditionalmathe ... is a mid point of bc. find (a) bc (b) bm (c) am 3. pqrstu is...

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