Question 1

Let a = i - k , b =x i + j +(1-x) k and c =y i +x j +(1+x-y) k . Then, [ a b ~ c ] depends on

Accepted Answer

Here, [a b c]=|a (b c)| Now, according to question, aligned [ array lll a & b c array ] & = | array ccc 1 & 0 & -1 \ x & 1 & 1 - x \ y & x & 1+x-y array | \ & =1(1+x-y-x(1-x))-1 (x^2-y ) \ & =1+x-y-x+x^2-x^2+y \ & =1 \ & = Independent of x and y . aligned

Question 2

Let ABC be a triangle and be a , b , c the position vectors of A , B , C respectively. Let D divide B C in the ratio 3: 1 internally and E divide AD in the ratio 4: 1 internally. Let BE meet AC in F .

Accepted Answer

Here we are given that OA = a , OB = b , OC = c Now P.V of Di.e OD = 1 OB +3 OC 1+3 aligned & OD = b +3 c 4 \ & OE = 4 OD + OA 4+1 = 4( ~b +3 c ) 4 + a 5 \ & OE = a + b +3 c 5 \ & Now, OE = 2 OB +3 OF 2+3 \ & OF = 5 OE -2 OB 3 \ & OF = 5( a + b +3 c ) 5 -2 ~b 3 \ & OF = a - b +3 c 3 P . V . of F is a - b +3 c 3 aligned

Question 3

If a =2 i + j +2 k , then the value of | i ( a i )|^2+| j ( a j )|^2+| k ( a k )|^2 is equal to

Accepted Answer

i ( a i )=( i i ) a -( i a ) i = j +2 k aligned & Similarly, j ( a j )=2 i +2 k \ & k ( a k )=2 i + j \ & \| j +2 k |^2+|2 i +2 k |^2+|2 i + j |^2 \ & =5+8+5=18 aligned

Question 4

A vector perpendicular to the plane containing the vectors i -2 j - k and 3 i -2 j - k is inclined to the vector i + j + k at an angle

Accepted Answer

A vector perpendicular to the plane is aligned & ( i -2 j - k ) (3 i -2 j - k ) \ & = | array ccc i & j & k \ 1 & -2 & -1 \ 3 & -2 & -1 array |=-2 j +4 k \ & unit vector a = -2 j +4 k 4+16 = -2 j +4 k 2 5 aligned Angle between the unit vector and r = i + j + k = ^ -1 r a | r | | a | = ^ -1 1 15 = ^ -1 14

Question 5

If a = i + j + k , a b =1 and a b = j - k , then b is

Accepted Answer

Given a = i + j + k , a b =1 and a b = j - k , Let b =b_1 i +b_2 j +b_3 k Now, j - k = a b = | array ccc i & j & k \ 1 & 1 & 1 \ b_2 & b_2 & b_3 array | b_3-b_2=0, b_1-b_3=1, b_2-b_1=-1 b_3=b_2, b_1=b_2+1 Also, a b =1 b_1+b_2+b_3=1 3 b_2+1=1 b_2=0 b_1=1, b_3=0 Thus, b = i

Question 6

107. u and v are two non-collinear unit vectors such that | u + v 2 + u v |=1. Then the value of | u v | is equal to

Accepted Answer

Question 7

If a is a vector of magnitude 50 , collinear with the vector b =6 i -8 j - 15 2 k and makes an acute angle with the positive direction of Z - axis, then a is equal to

Accepted Answer

Since a =m b for some scalar m i.e., array l a =m (6 i -8 j - 15 2 k ) \ |a|=|m| 36+64+ 225 4 \ 50= 25 2 |m| \ |m|=4 \ m= 4 array Since, a makes an acute angle with the positive direction of Z - axis, so its z component must be positive and hence, m must be -4. a=-4 (6 i +8 j - 15 2 k )=-24 i +32 j +30 k

Vector Algebra — Previous Year Questions & Solutions