Download VECTORS :- (1 mark questions)

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VECTOR ALGEBRA
SECTION - A (1 mark)
1. For what values of ‘λ’, the vectors (2î - 3 ĵ ) and (λî - 6ĵ) are parallel ?
2. Find the position vector of the centroid of a ΔABC where a , b and c are the position vectors of the
vertices A, B and C respectively.
3. Find position vectors of the points which divides the join of the points 2a - 3b and 3a - 2b externally
in the ratio 2:3.
4. Find the projection of the vector î +3 ĵ +7k on the vector 7î – ĵ + 8k.
5. Find the area of the parallelogram whose diagonals are 2î - 3ĵ + 4k and -3î + 4ĵ – k.
6. Find the cosine of an acute angle between the vectors 2î - 3ĵ + k and î + ĵ -2 k.
7. If | a + b | = | a - b | then find the angle between a and b .
8. If | a | = 5, | b | = 13 and | a x b | = 25. Find a . b .
9. Find the position vector of the mid point of the vector joining the points P(2,3,4) and Q(4,1,-2).
10. Find the value of ‘x’ for which x(î + ĵ + k ) is a unit vector.
11. If the position vector a of the point (5,n) is such that | a | = 13, find the value of n.
12. If the vector a = 2î - 3ĵ and b = -6î + mĵ are collinear , find the value of m.
13. If a vector makes angles α, β, γ with x-axis, y-axis, z-axis respectively, then what is the value of
sin2α + sin2β + sin2γ .
14. If | a + b |2 = | a |2 + | b | 2 , what is the angle between a and b ?
15. If a is a unit vector and ( x - a ).( x + a ) = 8 , then find | x |.
SECTION - B(4 Marks)
16. If a unit vector a makes angles π with î , π with ĵ and an acute angle θ with k , then find ‘θ’
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4
and hence the components of a .
17. Let a , b and c be the three vectors such that | a | = 3, | b | = 4, | c | = 5and each one of them being
perpendicular to the sum of the other two, find |a + b+ c |.
18. If with reference to right handed system of mutually perpendicular unit vectors î, ĵ and k, α = 3î – ĵ, β
= 2î + ĵ-3 k, then express β in the form of β = β 1 + β 2 where β 1 is parallel to α and β 2 is
perpendicular to α .
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19. Let a = î + 4ĵ +2k , b = 3î - 2ĵ + 7k and c = 2î – ĵ + 4k . Find a vector d which is perpendicular to
both a and b and c . d = 15.
20. If a , b and c are the position vectors of the vertices A,B and C of a ΔABC. Show that the area of the
ΔABC is 1 |a x b + b x c + c x a |. Also find the condition of the collinearity of these points.
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21. If a , b , and c represents the vectors BC, CA and AB of a ΔABC, then show that
a x b= b x c= c x a
Hence deduce sine formula for a triangle.
22. If a , b and c are vectors such that a . b = a . c , a x b = a x c and |a | = 0 then prove that b = c .
23. Three vertices of a triangle are A(0,-1,-2) , B(3,1,4) and C(5,7,1). Show that it is a right angled
triangle. Also find the other two angles.
24. If a x b = c x d
and b = c.
and
a x c = b x d , show that
a - d is parallel to b - c where a = d
25. If a , b , and c are three mutually perpendicular vectors of equal magnitude, prove that a + b +c is
equally inclined with vectors a , b and c.
26. Show that the angle between two diagonals of a cube is cos-1 1 .
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27. If a , b and c are three non zero vectors such that a x b = c and b x c = a
are mutually at right angles and | b | = 1 and | c | = | a |.
prove that a , b , c
28. If a , b and c be unit vectors such that a . b = a . c = 0 and the angle between b and c is
π , prove that a = ±2( b x c ).
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29. If the sum of two unit vectors is a unit vector, show that magnitude of their difference is √3
.
30. If a and b are unit vectors and θ is the angle between them, then show that sin θ = 1 | a - b |.
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ANSWERS
VECTOR ALGEBRA
1) λ = 4.
2) 1 ( a + b + c )
3
3) -5b
4) 60 units
√114
5) 3 √30
2
6) cos-1
7) 90º
8)60
9) 3î + 2ĵ + k
11) n = ± 12
12) m = 9
10) ± 1
sq. units
3
2√21
3
√3
13) 2
14) 90º
16) π ; 1 , 1 , 1
3 2 √2 2
17) 5√2
18) β 1 = 3 î – 1 ĵ and
2 2
β 2 = 1 î + 3 ĵ – 3k
2
2
20) b x c + c x a + a x b = 0
15) 3
19) 1(160î - 5 ĵ +70 k )
3
23) 45º , 45º