CHAPTER 4a Vector

SCKung M.Sc (Teaching Maths) B. Edu (Acct & Maths) FORM 5 ADDITIONAL MATHEMATICS TEL: 012 5349153

111Equation Chapter 1 Section 1CHAPTER 4: Vectors

4.1 Vectors

A vector is a quantity that has magnitude and direction.

Example:

is a vector whose magnitude is the same as the length of AB and whose direction is

from A to B.

Vector can be denoted as and its magnitude is written as or .

Zero or null vector is a vector with zero magnitude and is denoted as

Negative vector of is a vector with the same magnitude as but in a direction

opposite to , that is,

Two vectors are equal if they have the same magnitude and direction.

The product of vector by a scalar k is a vector whose magnitude is k times the

magnitude of , and is written as k

i) k is in the same direction as if k is positive

ii) k is in the a direction the opposite to if k is negative.

Example 1

Based on the diagram above, express the vectors , and in terms of .

Solution:

1


Two vectors are parallel if one vector is a scalar multiple of the other vector and vice

versa

i) is parallel to , if = λ , λ is a constant.

ii) = λ , if is parallel to

Example 2

Based on the diagram above, determine whether vector is parallel to each of the following vectors.

a) b) c) Solution:

Example 3

Given that and , determine whether vectors and are parallel.

2


If where λ is a constant, then the points A, B and C are collinear.

Example 4

a) Given that = 4 and 7 , show that points M, N and P are collinear.

b) Given that = 3 and 5 , show that points A, B and C are collinear.

If where and are non zero vectors and not parallel, then h = k = 0

Example 5

a) Given that where λ and β are constants, find the values of λ

and β if vectors and non -zero and non- parallel.

b) Given that where h and k are constants, find the

values of h and k if vectors and non -zero and non- parallel.

3

A

B

C


Example 6

Vectors and are non-parallel and non-zero. Given that

where m and n are constants, find the values of m and n.

Practice 4.1

1.

The diagram above shows eight vectors, , , , , , , , . Determine

whether each of the following pairs of vectors are equal.

a) and b) and

c) and d) and

4


e) and f) and

2.

In the diagram above, PQRS is a parallelogram. PR and SQ are the diagonals of the

parallelogram which is intersect at point T. State the vector which is equal to each of the

following vectors.

3.

Based on the diagram above, express each of the following vectors in terms of .

4. a) Given that and , determine whether vectors and are

parallel.

5


b) Given that , and ,determine whether vectors

and are parallel.

5. Given that and , show that points P, Q and R are collinear.

6. P, Q and R are three collinear points. Given that and point Q divides PR

internally in the ratio 2 : 3, express each of the following vectors in terms of

7. Given that where m and n are constants, find the value of m and n if

vectors and are non-parallel and non-zero.

8. Given and are two non-parallel and non-zero vectors. Find the values of h and k if

.

9. Vectors and are non-parallel and non-zero. Given that where u

and v are constants, find the values of u and v.

10. Vectors and are non-zero and non parallel. Given that

where h and k are constants, find the values of h and k.

6


4.2 Addition and Subtraction of Vectors

Resultant vector of two parallel vectors

when two or more vectors are combined and represented by a single vector, the single

vector is known as the resultant vector such as

Example 7

Given that = and , find

a) vector in terms of b) if

Example 8

In the diagram above, PQRS is a trapezium. PQ is parallel to SR. Given that PQ = SR,

find

a) vector in terms of b)

7


Resultant vector of two non-parallel vectors

Example 9

In the diagram above, PQR is a triangle, = , and = .a) State the vector which is equal to vector

. b) Find the resultant vector of vectors

and in terms of

8


Example 10

In the diagram above, PQRS is a parallelogram. Given that = and = , determine

a) the resultant vector of vectors and

=

b) vector in terms of and

Resultant vector of three or more vectors

Polygon Law

Example 11

9


The diagram above shows a heptagon PORSTUV. Find each of the following resultant vectors.

a) = b)

Subtracting two vectors

the subtraction of vectors that is is the addition of vector and the negative vector

of .∴ =

Example 12

In the diagram above, and are two parallel vectors. and =

a) If ,find vector in

terms of and hence mark point Q.

b) If find vector in

terms of and hence mark point M.

Example 13

10


In the diagram above, PQRS is a trapezium. PQ is parallel to SR. Given that = ,

= and SR = 2PQ.

a) Determine each of the following vectors in terms of .

i) ii)

b) Find the vectors which is equal to vector

Example 14

In the diagram PQRS is a parallelogram. It is given that and . and

intersect each other at point M. Find each of the following vector in terms of and

a) b) c)

11


Example 15

In the diagram above, PTR and QSR are straight lines. Points S and T lie on QR and PR

respectively such that S is the midpoint of QR and 3PT = PR. Given that

a)

b)

c) d)

12


Example 16

In the diagram above, , and S is a midpoint of . Given that point R and

point T are on the line and line respectively such that and .

i)Express each of the vector in terms of and .

a) b)

c) d)

13


ii) Prove that P, T and S are collinear and determine the ratio

Practice 4.2A

1. Find each of the following resultant vectors.

a) b)

2. Given that = and = find

a) vector 5 in terms of

b) if

3. Given OABC is a rectangle, where OA = 4 units and OC = 3 units. If and

, find

a) b)

4. Given A, B and C are collinear, where = . If

and find the value of k.

5.

14


In the diagram above, points P, Q and R are collinear. Given that = and

PQ : QR = 1 : 3, find vector in terms of . Hence, find if

6. A, B and C are three collinear points such that point B lies between points A and C, and

AB: BC = 1 : 4. Given that

a)

b)

c)

7.

The diagram above shows a triangle PQR. TS is parallel to PQ such that 5TS=3PQ. Given

that = , find the resultant vector of vectors and in terms of .

8.

15


In the diagram above, PQR is a triangle and S is the midpoint of PR. = and =.

a) State the vector which is equal to vector

b) Find each of the following vectors in terms of

i) ii)

9.

In the diagram above, PQRS is a trapezium. SR is parallel to PQ and point T lies on PQ

such that PT : TQ = 1 : 3, = and = .

a) Determine each of the following vectors in terms of .

i) ii)

b) If 2SR = PQ, find the resultant vector of vectors and in terms of

10.

In the diagram above, PQST and PQRS are parallelograms, = and =

a) State the vector which is equal to vector .

b) Determine the resultant vector of vectors and in terms of

11.

16


In the diagram above, PQRSTU is a hexagon. State each of the following resultant vectors.

a) b)

c)

12.

In the diagram above, PQRS is a parallelogram. T is the midpoint of SR. Given that

= and = , determine each of the following vectors in terms of .

a) b)

c)

13.

In the above diagram, = , = and BX : XA = 3 : 4. Find in terms of

.

14.

17


In the above diagram, CP: PA = 3 : 1 and the line PQ is parallel to the line AB. M is the

midpoint of the line PQ. If = and = express in terms of .

15. The diagram below shows ∆ABC. E is the midpoint of AB and D is a point on CB such

that CD = CB.

Given that = and = . Find

a) b)

16.

The diagram above shows a triangle PQR. Point S lies on RQ such that 4SQ = 5RS.

Given that = and = , determine each of the following vectors in terms of

.

18


a) b) c)

17.

In the diagram above, PQRS is a trapezium and PTRS is a parallelogram. T is a midpoint

of PQ. Given that = and = , express each of the following vectors in

terms of .

a) b) c)

18.

In the diagram above, PQR is a triangle. Point S lies on PQ such that 4PS = SQ and point

T lies on RQ such that 2RT = 3TQ. Given that = and = , find each of the

following vectors in terms of

a) b) c) d)

19.

19

SCKung M.Sc (Teaching Maths) B. Edu (Acct & Maths) FORM 5 ADDITIONAL MATHEMATICS TEL: 012 5349153 In the diagram above, PQRS is a trapezium. T and U are points lying on SR and PS

respectively, such that T is the midpoint of SR and . Given that = ,

= and , express each of the following vectors in terms of

a) b) c) d)

20.

In the diagram above, , , , and

. Express each of the following vectors in terms of and .

a) b) c) d)

Then show that A, T and S are collinear and determine

Example 17

20


In the above diagram E is a point on the line BD such that . The line AB is parallel

to the line DC, , and

a) Express each of the following vectors in terms of and .

i) ii) iii)

b) Hence, prove that BC is parallel to AE.

Example 18

21


In the above diagram, N is the midpoint of the line RT, , , and

.

a) Express each of the following vectors in terms of and

i) ii)

b) Determine whether the points P, N and S are collinear.

Example 19

22


In the diagram above, OSR and PSQ are two straight lines. = , =

PS : SQ = 2 : 3, and

a) Express vector in terms of

i) m, ii) n,

b) Hence, find the value of m and n.

Example 20

23


In the above diagram, = and = . OA is produced to C such that OC = 3OA. OB is produced to D such that OD =2OB.The lines AD and BC intersect at P.

a) Find each of the following vectors in terms of

a) b)

b) Given that and , express in terms of

a) m,

b) n,

c) Hence, find the value of m and of n.

24

SCKung M.Sc (Teaching Maths) B. Edu (Acct & Maths) FORM 5 ADDITIONAL MATHEMATICS TEL: 012 5349153 Practice 4.2b1.

In the diagram, , is parallel to and . Given that

and , express each of the following vectors in terms of and .

a) b) c) Hence, prove that PS is parallel to QR.

2. In the diagram, below, ABCD is a trapezium and E is the midpoint of AC.

It is given that , and , where k is a constant.

a) Find in terms of and .

b) Find in terms of k, and . Hence, find the value of k is BE is parallel to CD.

3. In the diagram, M and N are midpoints of OP and MQ respectively. Given that

, and , find each of the following vectors in terms of and .

a) b) Hence, prove that the points R, N and P are collinear.

25


4.

In the diagram, ABC is a straight line such that . The point T on the line BD is

such that . The point S on the line DC is such that . It is given that

and .a) Express

i) ii)

in terms of and b) Determine whether the point A, T and S are collinear.

5.

In the diagram above, OPQ is a triangle. OTR and PTS are two straight lines. = ,

= , PR : RQ = 1 : 2 and 3SQ = OS.

a) Express each of the following vectors in terms of

i) ii) iii)

b) If OT = mOR and PT = nPS, express vector in terms of

i) m, ii) n,

c) Hence, find the values of m and n .

26


6.

In the diagram above, OPQR is a parallelogram. OUS and PUT are two straight lines. The

length of RS is three times the length of SQ and T is the midpoint of OR. = and

= .


i) ii) iii) iv)

b) Given that OU =hOS and PU =kPT, express vector in terms of

i) h, ii) k,

c) Hence, find the value of h and k.

d) If the area of ∆OPU = 36 unit2, find the area of ∆OPS

7. In the diagram, = , = and = .


i) ii)

b) Given that and , express in terms of

27


i) m, ii) n,

c) Hence, find the value of m and n.8.

In the diagram, ABCD is a quadrilateral such that the line DB intersects the line XC at Y.

It is given that = , , = and = .

a) If and , find the value of m and of n.

b) Hence, find AB : XY.

9.

In the diagram, = and = . The point C lies on OA such that OC: CA = 2 : 1 while the point D lies on OB such that OD : OB = 2 : 3 . The straight lines AD and BC

intersect at point E. It is given that and , where h and k are constant.

a) Express in terms of

i) h, ii) k,

b) Hence, calculate the value of h and of k.

28


10. In the diagram below, = , = and OP = OB. M is the midpoint of AB.


i) ii)

b) Given that , express in terms of p,

c) Given that , express in terms of k, .

d) Hence, find the value of p and k.

11. The diagram below shows a triangle OXY. The straight line AY intercepts the straight

line BX at C. It is given that = , = , OA= OX and OB = BY.

a) Express each of the following vectors in terms of .

i) ii) iii)

29


b) Given that and , find the value of h and of k.

12. The diagram below shows a triangle OAB such that and = . Point C lies on the straight line AB such that AC: CB = 2 : 1 and D is the midpoint of OB.

a) Find each of the following vectors in terms of :

i) ii)

b) OC is produced to point E such that DA is parallel to BE. Given that = h and

, express in terms of

i) h, ii) k,

c) Hence, find the value of h and k.

13.

In the diagram above, OABC is a parallelogram with and . ABR is a

straight line with . Express each of the vectors in terms of .

a) b)

30


Given also and , express

c) in terms of and

d) in terms of , and . Hence, find the value of and

4.3 Vectors in a Cartesian Plane

Vectors in the form

In Cartesian plane, vector → magnitude of 1 unit towards the positive direction of x- axis

vector → magnitude of 1 unit towards the positive direction of y- axis

Vectors and are known as unit vectors in the horizontal and the vertical directions respectively.

In the diagram, and 3

=

The vector in a Cartesian Plane which proceeds from point A to point B is:

31


Example 21

The diagram above shows three vectors, , and , in a Cartesian plane. Express

each of these vectors in the form .

Example 22a) Given points P(5, 1) and Q(−2, 4), express

vector in the form

b) The coordinates of A and B are (−3, 4) and (2, −1) respectively. O is the origin. Express each of the following in the form

.

i) ii)

32


Vector in the form

Vector in a Cartesian plane can also be expressed in the form

The vector in a Cartesian Plane which proceeds from point A to point B is:

Example 23

For each of the following, express vector in the form a)

i) ii) P = (4, −6) and Q = ( −3, 7) b) Given and

Magnitudes of Vectors

33


Example 24Determine the magnitude of each of the following vectors.

a)

b)

Unit vectors in given direction

Vector is in the direction of vector and has a magnitude of 1 unit. Vector is

known as the unit vector in the direction of

Unit vector in the direction of is

If , unit vector in the given direction of is

Example 25a) Determine the unit vector in the direction

of if b) Given that , find

34


c) Find the unit vector in the direction of

d) Given . Find the unit vector in

the direction of

Adding and Subtracting two or more vectors

Example 26

a) Given that , , and , find

i) ii)

b) Given that , , and , find

i) ii)

c)

35


i) Given that find

ii) Given that and , find

Multiplying vectors by scalars

Example 27

a) Given that , find

i) ii)

b) Given that , find

i) ii)

36


Combined operations on Vectors

Example 28

a) Given that and , find

i) ii)

b) Given that find

i) ii) the unit vector in the direction of

Example 29

a) Given and . Calculate the values of p if

i) ii) are parallel

37


b) If and , find the value of n if vectors are parallel.

Practice 4.31.

In the diagram above, vectors are drawn in a Cartesian plane.

Express each of these vectors in the form .

2. For each of the following pairs of points P and Q, express vector in the form . a) P(3, 4) Q(7, 9) b) P(−2, 3) Q(2, −7) c) P(5, −1) Q(−9, 2) d) P(−6, −3) Q(−1, −6)

3.

38


The diagram above shows vectors, in a Cartesian plane. Express each

of these vectors in the form .

4. Express each of the following vectors in the form .

a)

b) such that P = (−4, −6) and Q = (9, −1)

c)

d) such that U is the origin and V = (−3, 5)

5. Determine the magnitude of each of the following vectors.

a) b)

c) d)

e)

f)

6.

39


In the diagram above, vectors are drawn in a Cartesian plane. Determine the magnitude of each of these vectors.

7. For each of the following vectors, determine the unit vector in the direction of the vector.

a) b)

c) d)

e) f)

8. Given that and , find

a) b) c)

9. Given that , , and , find

a) b) c)

10.

40


In the diagram above, OPQR is parallelogram. Given that and

, determine vector .

11. Given that , and , find

a) b) c)


a) b) c)

13. Given that , and , determine each of the following vectors.

a) b)

14. Given that , find

a) b) c)

15. Given that , find

a) b) c)


a)

b)

c) the unit vector in the direction of

17. Given that , and , determine

a)

41


b) the magnitude of vector

c) the unit vector in the direction of

18.

In the diagram above, OAB is a triangle. and point C lies on AB such that AC : CB = 2 : 3

19. In the Cartesian plane with O as the origin, and

.a) Determine each of the following vectors.

i) ii) iii) iv)

b) Hence, find the unit vector in the direction of

20. If and , determine

a) the values of m and n if

b) the vector that is parallel to vector and has a magnitude of 15 units.

21. Given that and , find the possible values of h if

a) unitsb) points P, Q and R are collinear

22. Given that O is the origin, , and , find

42


a) the coordinates of point P

b) the unit vector in the direction of

c) the value of m if vector is parallel to vector

23. PQRS is a parallelogram. and where h and k are constants.a) Find the value of h and k b) Calculate the length of the diagonal PR

24. It is given that and where p is a constant. Find the possible value of p if

a) b) the points A, B and C are collinear.

25. It is given that P(1, −3) and Q(5, 2). If , where h and k are constants, finda) the value of h and k


26. If and , find the value of m if is parallel to the x-axis.

27. Given that and , find the value of p if is parallel to

28. It is given that and . The coordinates of the points P and Q are

(2, −4) and (8, 2) respectively. Given that , where k is a constant, find a) the value of m and k


Answer:

43

SCKung M.Sc (Teaching Maths) B. Edu (Acct & Maths) FORM 5 ADDITIONAL MATHEMATICS TEL: 012 5349153 Practice 4.11a) Yes b) No c) Yes d) No e) Yes f) Yes

2a) b)

c) d)

3a) b)

c) d) 4a) Yes b) No

6a) b)

7. m = , n = 8. h = 2, k = 2; h = −2. k = 6

9. u = −2, v =

10. h = ,k =

Practice 4.2A

1a) b)

2a) b) 180 units

3a) b) 3 units4. k = 9

5. , units

6.a) b) c)

7.

8a)

b)i) ii)

9a)i) ii)

b)

10a) b)

11a) b) c)

12a) b)

c)

13.

14.

c)

17a) b)

c)

18a) b)

c) d)

19a) b)

c) d)

20a) b)

c) d)

Practice 4.2B

1.a) b)

c)

2.a)

b)

3.a) b)

R,N and P are collinear

4.a)i) ii)

b) A, T and S are collinear

5.a) i) ii)

iii)

b) i) ii)

44


15a) b)

16a) b)

iii) iv)

b) i) ii)

c) k = ,h = d) 99 unit2

7a)i) ii)

b)i) ii)

c)

8a) m= , n = b) 5 : 2

9a) i)

ii)

b) , k =

10a) i ) ii)

b) c)

d) k = , p =

11a)i) ii)

iii) b) , k =

12a) i ) ii)

b) i) ii)

c) k =

13) a) b)

c) d)

Pactice 4.3

c) , n =

6a)i) ii)

3a) , ,

,

4a) b)

c) d) 5a) 7.21 units b) 8.54 units c) 9.06 units d) 8.60 units e)9.22 units f) 4.47 units

6. units, units,

units, units

7a) b)

c) d)

e) f)

8a) b) c)

9a) b) c)

10.

11a) b) c)

12a) b) c)

13a) b)

45


1.

2a) b)

c) d)

15a) b) c)

16a) b) units

c)

17a) b) units c)

18a) i) ii)

iii) b)

19a) i) ii)

iii) iv)

b)

20a) m = b) 21a) 2, 26 b) 77

22a) (−5, −3) b) c)

23a) h = 7, k = b) 8.32224a) p= 25 or 7 b) p= −16

14a) b)

c)

46


25a) h = , k = 3 b) 26. m = 327. p = 1

28a) m = , k = b)

47

CHAPTER 4a Vector

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