CHAPTER 9 Baru 1 Tutorial and All Questions

8/2/2019 CHAPTER 9 Baru 1 Tutorial and All Questions

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Chapter 9 Vectors

40

TUTORIAL CORNER

1. Vector location for point A and B relative to origin O are a and b.

OA is lengthen to C where OC = 3OA and OB are lengthen to D

where OD = 2OB. Lines AD and BC intersect at P. Express AD and

BC in term of a and b.

Given AP = AD and BP = BC. Express OP

a) In term of , a and bb) In term of , a and b

Determine the value for and . Then, get the location vector for P

2. Points A and B have location vector a and b. Point C situated at AB

with condition . Point D is middle point for OC. Line AD

that lengthen will meet OB at E. Find in term of a and b

a) OCb) vector AD

3. Given a = 2i – 5j and b = 3i + 2j. Find

a) Unit vector in direction ab) Unit vector in direction bc) a – bd) 2a – 4b

4. Find cosine angle between a and b

a) a = 5i – 12j and b = 12i + 3j

b) a = <8,2,7> and b = <8,5,9>

5. Determine the value of a if vector 5i + 13j – 2k and vector7ai – 4j + 5k are parallel.

6. Find the vector product a x b if a = 2j + 4k and b = 10i + 3j + 8k

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Chapter 9 Vectors

41

7. Given that vectors A, B and C area = i + j + kb = i + 2j + 3kc = i – 3j + 2k

Find

a) unit vector that parallel with a + b + cb) cosine angle between vector a + b + c and vector ac) vector in type i + j + k that perpendicular with a and bd) vector location for D where ABCD is a parallelogram and BD is

the diagonal.

8. Refer to the origin O, vector location for points A, B and C are

= 9i + 7j –

k= 3i – 11j + 5k

= 5i – 5j – k

a) Find the cosine angle CAB and prove the triangle area ABC =

b) Find the vector D and E where D is a coordinate dividing ABinto three segments and is situated near to A. E is midpoint of CD.

9. If p = 6i + 2j – k, q = 9i – 4j – 6k, r = 2i – 8j + 5k. Find

a) p x qb) r x qc) (p x q) x rd) p x (q x r)



Chapter 9 Vectors

42

PAST YEAR QUESTIONS CORNER

1. (a) Given vectors u = 2i + 5j and v = 4i 7j and w = 3i + 6j.

Express these below in term of i and j.

(i) 6u 5v 4w(ii) u + 5v 7w

(b) Given u = < 4, 3, 4 >, v = < 8, 6, 3 > and w = < 7, 5, 1>. Find the

value for these expression:

(i) u . v (ii) u . ( v . w ) (iii) v . ( u + v)

2. (a) Given vectors u = 3i – 4j and v = i + j. Express this vector w = 2u – v

in term of i and j. Then, determine the unit vector unit in direction w.

(b) Given vector a = 2i + 3j and vector b = i + 5j – 3u . Find

i) a bii) b a

3. Vector locations for A, B and C are

, ,

Calculate the value for m and n if

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

(a) vector 2a – b (b) unit vector in the direction of vector 2a – b.

5. Given a = 2i + 4j – k

b = 3i –

2j +4k

Find :

a) a x b

b) (a – b) x (a + b



Chapter 9 Vectors

43

BANK QUESTIONS CORNER

1. In above figure, Z is a point on XY so that XZ = ZY

a) State in term of and

b) State in term of and

c) Prove that

2) The above figure show a triangle ABC with M as middle point to OA andP located at AB with condition AP = 3PB. Middle point for OP is N. If given

and

a) i) Calculate vector and in term of p and q

ii) Given OC k q and line MN meet OB at C. Get the valuefor k

b) If coordinate P is (4,3) and = 2i + 5j, get the coordinate for Q. Find the unit vector unit for Q and determine the for vector PQ.

O

YX Z

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B

A

C

O

M

P

N



Chapter 9 Vectors

44

3. Location vectors for three points A, B, C are 2a + b, 4a – 2b and 8a – 8b.Prove that these three points are located on a straight-line and find theratio AB: BC

4. ABCD is a quadrilateral and P, Q, R, S are middle points for sides AB ,BC, CD and DA . Prove that PQRS will make a parallelogram.

5. Find the magnitude for vector i – 2j +2k and find unit vector unit thatparallel with vector i – 2j + 2k.

6. If a = i – j – 2k, b = 2i + j , c = -2i + j 2k, find unit vector in direction a and

2b + c.

7. ABC has coordinates (2,3,4), (-2,1,,0) and (4,0,2). Find an acute angle forBAC.

8. There is a triangle ABC with vertexes A(2,3,4), B (0.1.2) and C (2,0,-1).Find unit vector that perpendicular with plane that have A,B,C .

9. If a = i + j + k, b = 3i – 4j + 5k, show that

i) a x b = b x a

ii) a . a x b = c

10. Vector location for points A, B, C are a = i – 2j + k, b = 2i – 3j – k, c = 3i – j + 2k, Find

i) ii) angle BAC

A B

CD

P

Q

R

S

B

O

C

A



Chapter 9 Vectors

45

11. Let O is an origin, P is point (2,4) and Q is point (1,3). Given vector

. Get

i) coordinate R

ii)

i) Given S is points (4,7) and where h and k areconstant, calculate values for h and k

12. Vectors location for A,B and C are OA = 2i – 3j, OB = 3i+ 4j

and = 6j. Get

i)

ii)

iii) Values for m and n if

13. OPQR is a parallelogram with = 12i – 5j and = 4i + 3j. Find

i) ii) unit vector iniii) angle OPQ

14. i) Show that 2i – j + 4k and 5i + 2j 2k are perpendicularii) Find the third vector that perpendicular with both vectors above

15. i) Find unit vector that perpendicular to plane that have vectors.a = i – 2j + k and b = -2i + j+ 2k

ii) Angle between two vectors a with b are cos-1 . Find the value q, if

a = 6i –

8j b = 4i + qj

16. Vector locations for points A, B and C are

a = 4i – 9j – k, b = i + 3j + 5k. c = pi – j + 3k



Chapter 9 Vectors

46

a) Find unit vector unit that parallel with vectorb) Find p value so that A, B and C are a straight line c) If p = 2, find vector location for D so that ABCD is a

parallelogram.

17. Figure above show an equivalent hexagon that centered at O . If = a and

= b, express these vectors in simple way.

i) ii) iii) iv)

18. In the above figure, SP = QR = RS. If OP AND OS arep = 2i + j and s = 3i – 5j

i) express and in term of i and

ii) find and

KJ

L

GH

IO

P

Q

S

O

R



Chapter 9 Vectors

47

19. Vector , , and are 4, 6, 2b, and 6b. Point K dividing FC with the ratio of 1: 6.

i) State and in term of a and bii) Prove that K located on the BE line

iii) Get the values for ratio

20. In the above figure, OA = a and OB = b. Point P and Q located at AB and OB with condition 3 AP = PB and 4OQ = 3OB. State

i) and in term of a and b

ii) If and , show that m = 3n and 3m + 4n

iii) Solve that equations and get the ratio values for

A

O B

P

Q

S



Chapter 9 Vectors

48

ANSWERS FOR BANK QUESTIONS CORNER

1. a) b)

2. a) i) = 6q –

4p , , ii) k = 3

Q = (2,2), Direction PQ = 111 48 with x-axis and in an anti-clockwise.

3. AB :BC = 1 : 2

5. 3,

6. ,

7. BAC = 7558

8. 10. i) i – 5j + 3k ii) BAC = 8024

11. i) Coordinate R(3,1) ii) 4i – 2j iii) h = , k = 3

12. i) ii) i – 7j iii) m = , n =

13. i) 16i – 2j ii) iii) OPQ = 12031

14. ii) 6i + 24j + 9k

15. i) 5i – 4j + 5k

ii) q = 3,

16. a) b) p = 2 c) i – 13j – 3k

17. i) (a + b) ii) a iii) 2a iv)b

18. i) , ii) ,

19. i) ,

20. i) , iii) AS : AQ = 4 : 13

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these questions.

It will help you

improve your

skills on thisto ic. Good

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questions? Don’t give up

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Chapter 9 Vectors

49

ADDITIONAL QUESTIONS CORNER

1. In the figure below, OA = a and OB = b.

a)

Express OP in term a and b.b) Given that OQ = ma + nb with m and n is constant. Findvalue for m and n.

2. In figure below, PR = a and PQ = 2a + b.

a) Mark and labeli) point S with condition PS = bii) point T with condition PT = a + b.

b) Given that SQ = kST, find the value for k.

a

2a + b

3. Given AB = (h 2)a and AB = 5a, find the value for h

4. p and q are two vectors and | p | = 3. Find the value for | p + q | for everycase below :

i) q = 3p

ii) q = 3piii) q perpendicular with p and | q | = 4.

5. Given a = (h +1)i +6j and b = 6i – (k+2)j. If a = b, find value for h and k .

6. Given a = i + j, b = i + 2j and c = i – 3j. Find :i) | a + b + c |ii) unit vector that parallel with a + b + c.

Q

bO

a

B

A · P

P

R



Chapter 9 Vectors

50

7. Given = 3i + 8j, = 6i + 2j and = x i + y i. Find the value for x and y so

that .

8. Given a = 2i + k j and b = m i + 2j. If a and b are parallel, find the relation

between k and m .

9. Given a = 3i + j, b = 5i – 2j and c = mi – 6j. Given also u = b – a and v = c – b.

a) Find u and vb) If u and v are parallel, find the value for m c) Then, find | u | : | v |.

10. ABCD is a parallelogram and p is a point with DC. BC is lengthen to E so thatBC = CE, while BP is lengthen to meet at Q. Given = 4x and = 2y.

a) Express every vectors below in term of x and y .

i. ii.

iii. b) Given that BQ = h BP, express BQ in term of h , x and y .

c) Given that DQ = k DE, express BQ in term of k , x and y . Then,find the value for h in (b) and value for k .

C

Q

D

BA

E

2y

4x

P



Chapter 9 Vectors

51

ANSWERS FOR ADDITIONAL QUESTION CORNER

1. a) OP = 2a + 2b c) m = 1, n = 3

2. a) b) k = 2

3. 7

4. a) 12 b) 6 d) 5

5. h = 5, k = 8

6. a) 3 b) i

7. x = 12, y = 10

8. mk = 4

9. a) u = 2i – 3j b) c) 3 : 4

v = (m –

5)i + 4j

10. a) i. 2y – 4x b) BQ = 2hy – 2hx c) (4k – 4)x +(2k + 2)y

ii. 2y – 2x

iii. 4x + 2y

R

P

Q

T

S

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CHAPTER 9 Baru 1 Tutorial and All Questions