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)
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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
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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
Try solving these
questions. It can
help you improve
your skills in this
topic. Good luck!!
B
A
C
O
M
P
N
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Chapter 9 Vectors
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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
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Chapter 9 Vectors
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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
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Chapter 9 Vectors
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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
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Chapter 9 Vectors
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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
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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
Try solving
these questions.
It will help you
improve your
skills on thisto ic. Good
Can you solve these
questions? Don’t give up
Ask your tutor or your
friends if ou have
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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
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Chapter 9 Vectors
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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
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Chapter 9 Vectors
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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