Vector Algebra

Consider a ABC where A(1,3,2), B(-2,8,0) and C(3,6,7). If the angle bisector of BAC meets the line BC at D, then the length of the projection of the vector AD on the vector AC is :

1. 372 38
2. 19
3. 392 38
4. 382

Let a= i+ j+ k, b=- i-8 j+2 k and c=4 i+c2 j+c3 k be three vectors such that b a= c a. If the angle between the vector c and the vector 3 i+4 j+ k is , then the greatest integer less than or equal to 2 is .

Let a=3 i+ j-2 k, b=4 i+ j+7 k and c= i-3 j+4 k be three vectors. If a vector p satisfies p b= c b and p a=0, then p ( i- j- k) is equal to

1. 24
2. 32
3. 36
4. 28

Let a and b be two vectors such that | a|=1, | b|=4, and a b=2. If c=(2 a b)-3 b and the angle between b and c is , then 192 2 is equal to .

Let a=3 i+2 j+ k, b=2 i- j+3 k and c be a vector such that ( a+ b) c=2( a b)+24 j-6 k and ( a- b+ i) c=-3. Then | c|2 is equal to .

Let a=a1 i+a2 j+a3 k and b=b1 i+b2 j+b3 k be two vectors such that | a|=1, a b=2 and | b|=4. If c=2( a b)-3 b, then the angle between b and c is equal to:

1. -1 (- 1 3 )
2. -1 ( 23 )
3. -1 ( 2 3 )
4. -1 (- 32 )

Let A(2,3,5) and C(-3,4,-2) be opposite vertices of a parallelogram ABCD. If the diagonal BD= i+2 j+3 k, then the area of the parallelogram is equal to :

1. 12 410
2. 12 306
3. 12 586
4. 12 474

Let a= i+ j+ k, , R. Let a vector b be such that the angle between a and b is 4 and | b|2=6. If a b=3 2, then the value of ( 2+ 2 )| a b|2 is equal to

1. 85
2. 90
3. 75
4. 95

Let a and b be two vectors such that | b|=1 and | b a|=2. Then |( b a)- b|2 is equal to

1. 1
2. 3
3. 5
4. 4

Let a, b and c be three non-zero vectors such that b and c are non-collinear. If a+5 b is collinear with c, b+6 c is collinear with a and a+ b+ c= 0, then + is equal to

1. 30
2. -30
3. -25
4. 35