Vector Algebra. a) only x b)only y c)neither x nor y d) both x and y. 5)Two adjacent sides of a parallelogram PQRS are given by: PQ =2i +j +11k &

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Randell Chandler
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1 Vector Algebra 1)If [a хb b хc c хa] = λ[a b c ] then λ= a) b)0 c)1 d)-1 )If a and b are two unit vectors then the vector (a +b )х(a хb ) is parallel to the vector: a) a -b b)a +b c) a -b d)a +b )Let a =i -k b =xi + j + (1-x)k and c =yi +xj +(1+x-y)k then [a b c ] only depends on : a) only x b)only y c)neither x nor y d) both x and y 4) If the vectors ai +j +k, i +bj +k & i +j +ck (a b c 1) are coplanar, then the value of abc-(a+b+c) is a) b)0 c)-1 d)- 5)Two adjacent sides of a parallelogram PQRS are given by: PQ =i +j +11k & PS =-i +j +k &. The sides PS is rotated by an acute angle α in the plane of the parallelogram so that PS becomes PS. If PS makes a right angle with the side PQ, then the cosine of the angle α is given by a) 8 9 b) 17 9 c) 1 9 d) )Let a and b be two unit vectors. If the vectors c =a +b & d = 5a -4b are perpendicular to each other then the angle between a & b is a) π 6 b) π c) π d) π 4 7)The value of a so that the volume of parallelepiped formed by vectors i +aj +k, j +ak, ai +k becomes minimum is : a) b) c) 1 d)

2 8)The vectors a = i +j +(m+1)k, b = i +j +mk, c = i -j +mk are coplanar for: a)m = 1 b) m = 1 c)m= d)no value of m 9)If the vectors AB =i +4k & AC =5i -j +4k are the sides of a triangle ABC, then the length of the median through A is: a) 7 b) c) 45 d) 18 10) Let a, b, c be three non-zero vectors, which are pairwise non-collinear. If a +b is collinear with c and b +c is collinear with a, then a +b +6c is: a) a b)b c)o d) a +c 11) Let a, b, c be three non-coplanar vectors & p, q, r be vectors defined by the relations p= b хc [a b c ] q = c хa [a b c ] r = a хb [a b c ] then the value of the expression: (a +b).p +(b +c ). q +(c + a ). r is equal to: a)0 b)1 c) d) 1)If p, q & r are unit vectors satisfying p -q +r =o, then the angle between the vectors p and r is a) π 4 b) π c) π 6 d) π 1)If x =i -6j -k, y =i +4j k &z =i -4j -1k then the magnitude of the projection of x хy on z is a)1 b)15 c)14 d)1 14)If x, y & z be three unit vectors such that x (y х z )= (y +z ). If y is no parallel to z then the angle between x &y is a) π 4 b) π c) π d) 5π 6

3 15)Volume of parallelopiped determined by vectors a, b, c is. Then the volume of the parallelepiped determined by vetors (a хb ), ( b хc ) & (c хa ) is a)100 b)0 c)4 d)60 16) Let a, b, c be three non-zero vectors such that no two of them are collinear & (a хb ) хc = 1 b c a. If θ is the angle between vectors b & c then value of sin θ is a) b) c) d) 17)Direction of zero vector a)does not exits c)is indeterminate b) is towards origin d)none of these 18)If in a right angle triangle ACB, the hypotenuse AB=P then AB.AC + BC.BA +CA.CB is equal to a)p b) p c) p d)p 19)If a (b c )= then a) c (a b )=- b) a (c b )=- c) b (a c )= d)(a c ) b = 0)Let a =i +j -k b =i +j. If c is a vector such that ) a c = c, c a = and the angle between a b and c is 0 then (a b ) c equals a) 1 b) c) d) 1)OABC is a parallelogram with OC =a and AB=b then OA =.. a)a +b b)a -b c) 1 (b a ) d) 1 (a -b ) )If a + b = a b then

4 a) a and b are perpendicular to each other b) a is parallel to b c) a =0 d) b =0 )Let a =i +j +k, b =i -j +k andc =xi +(x )j -k, If the vector c lies in the plane of a and b then x equals a)1 b)-4 c)- d)0 4) The non zero vectors a, b & c are related bya =8 b and c =-7b. Then the angle between a and b is a)π b)0 c) π 4 d) π 5)ABCDEF is a regular hexagon. If AB =a & BC =b then CD =.. a)a +b b)b -c c)a -b d) c -b 6)Let u, v & w be vectors such that u + v + w =o. If u =, v =4 & w =5 then u v +v w + w u is a)47 b)-5 c)0 d)5 7) Let the vectors a, b, c & d be such that (a b ) ( c d )= o. Let P 1 &P be the planes determined by the pair of vectors a, b and c, d respectively. Then the angle between P 1 &P is a)0 b) π 4 c) π d) π 8)Let the unit vectors A and B be perpendicular and the unit vector C inclined at an angle θ to both A and B. if C =αa+βb+γ (A B ) then a)γ =1-α c) β = 1+cosθ b) γ =-cosθ d) all above

5 9)Consider the parallelepiped with sides a =i +j +k, b =i +j + k, c =i +j +k then angle between a and the plane containing the face determined by b and c is a)sin 1 1 b)cos c) sin d) sin 1 0)Let a =j -k and c =i -j -k then the vector b satisfying a b + c =o and a b = is a) i -j +k b) i -j -k c) i +j -k d)- i +j -k 1)For non-coplanar vectors A, B and C, (A B ) C = A B C holds iff a) A B = B C = C A =0 b) A B = B C =0 c) A B = C A =0 d) B C = C A =0 )If the unit vectors a and b are inclined at an angle θ and a b <1 then if 0 θ π,θ lies in the interval. a)[0, π ] b)( 5π 6, π] c)[ π 6, π ] d)( π, 5π 6 ] ) Let OA =a, OB =10a +b and OC =b, where O, A and C are non collinear points. Let p denote the area of the quadrilateral OABC and let q denotes the area of parallelogram with OA and OC as adjacent sides. If p=kq then k is a)6 b) c) d)8 4)The volume of the parallelepiped whose sides are given by OA = i -j OB = i +j -k, OC =i -k is 1) 4 1 b)4 c) 7 d)-4 5) If is the area of the triangle whose vertices A, B and C have position vectors i -j + k, i +j -k & i j +k w. r. t. origin of reference O then is a)6 b)± 1 c)1 d) 6

6 6)Given a = i + j -k, b = i + j + k & c = i + j -k. A unit vector perpendicular to both a + b and b + c is a) i +j +k 6 b) j c)k d) i +j +k 7)If a b = c and b c = a then a) a, b, c are orthogonal in pairs but a c b) a, b, c are orthogonal but b 1 c) a, b, c are orthogonal to each other d) a, b, c are orthogonal in pairs and a = c, b =1 8)If a & b are two unit vectors and is the angle between them, then 1 a b is equal to a)0 b) π c) sin d) cos 9)Projection of i + j +k on i j -k is equal to a) b)- c)9 d)-9 40) Let a = i +j -k, b =i +j. If c is a vector such that a c = c, c a = and the angle between a b and c is π then (a b ) c equals 6 a) 1 b) c) d)

CHAPTER 10 VECTORS POINTS TO REMEMBER

For more important questions visit : www4onocom CHAPTER 10 VECTORS POINTS TO REMEMBER A quantity that has magnitude as well as direction is called a vector It is denoted by a directed line segment Two