### Eamcet - Maths - Vector Algebra

Observe the following statements : A : Three vectors are coplanar if one of them is expressible as a linear combination ofthe other two. R : Any three coplanar vectors are linearly dependent.Then which of the following is true

A.  Both A and R are true and R is the correct explanation of A

B.  Both A and R are true but R is not the correct explanation of A

C.  A is true, but R is false

D.  A is false, but R is true

If |a+b|2=|a|2+|b|2 then the angle between a and b is

A.  900

B.  600

C.  450

D.  1200

A: Angle between the vector i-2j+k, 2i-j-k is π/3 R: If θ is  the angle between a, b then cosθ=a.b/|a||b|

A.  Both A and R are true and R is the correct explanation of A

B.  Both A and R are true but R is not correct explanation of A

C.  A is true R false

D.  A is false but R is true

If p=(2, 1, 3), q=(-2, 3, 1), r=(3, -2, 4) and j is the unit vector in the direction of y-axis then (2p+3q-4r). j=

A.  18

B.  19

C.  20

D.  21

If three points A, B, C have position vectors (1, x, 3) and (y, -2, -5) respectively and if they are collinear, then (x, y)=

A.  (2, -3)

B.  (-2, 3)

C.  (-2,-3)

D.  (2, 3)

I: If the vectors a=(1, x, -2), b=(x, 3, -4) are mutually perpendicular,then x=2 II: If a=i+2j+3k, b=-i+2j+k, c=3i+j and a+tb is perpendicular to c then t=5

A.  Only I is true

B.  Only II is true

C.  both I and II are true

D.  neither I nor II are true

If a=i+4j, b=2i-3j and c=5i+9j then c=

A.  2a+b

B.  a+2b

C.  a+3b

D.  3a+b

The vector area of the triangle whose adjacent sides i-2j+2k, 3i+2j-5k is

A.  1/2(6i+11j-8k)

B.  1/2(6i-11j+8k)

C.  1/2(6i+11j+8k)

D.  1/2(6i-11j-8k)

The vector c directed along the internal bisector of the angle between the vectors 2i+3j-6k and -2i-j+2k with |c|=√21 is

A.  ±(-8i+2j-4k)

B.  ±(-4i+j-2k)

C.  ±(-12i+3j-6k)

D.  none

The vector equation of the plane passing through the point 2i+2j-3k and parallel to the vectors 3i+3j-5k, i+2j+k is

A.  r=s(2i+j-k)+t(i+2j+2k)

B.  r=2i+2j-3k+s(3i+3j-5k)+t(i+2j+k)

C.  r=(i+2j+3k)+s(-2i+3j+k)+t(2i-3j+4k)

D.  none

A unit vector perpendicular to the plane of a=2i-6j-3k, b=4i+3j-k is

A.  4i+3j-k /√26

B.  2i-6j-3k/7

C.  3i-2j+6k/7

D.  2i-3j-6k/7

If a=(1, 1, 1), c=(0, 1, -1) are given vectors then a vector b sastisfying the equations axb=c and a.b=3 is

A.  5i+2j+2k

B.  5/2i+j+k

C.  5/3i+2/3j+2/3k

D.  i+2/5j+2/5k

The magnitude of the projection of the vector a = 4i - 3j + 2k on the line which makes equal angles with the coordinate axes is

A.  √2

B.  √3

C.  1/√3

D.  1/√2

If AB=2a+b and AD=a-2b where |a|=1, |b|=1 and (a, b)=600 are the adjacent sides of a parallelogram, then the length of the diagonal BD is

A.  √13

B.  √7

C.  √12

D.  none

Arrange the magnitudes of following vectors in ascending order A) ixj+ jxk+kxi  B) If lal=2, lbl=3, (a, b)=450 then axb C) (2i-3j+2k)x(3i-j+4k)

A.  A, B, C

B.  C. B, A

C.  B, C, A

D.  B, A, C

The vector of magnitude √51 which makes equal angles with the vector a=1/3(i-2j+2k), b=1/5(-4i-3k), c=j is

A.  ±(i+2j-k)

B.  ±(2i+j-k)

C.  ±(5i-j-5k)

D.  none

Statement I: The points 4i+5i+k, -j-k, 3i+9j+ 4k and -4i+4j+4k are coplanar Statement II  : The given points from  the  vertices of a parallelogram. Which of the following is true? a)  Both statements  are  true and statement II is correct explanation of statement I b)  Both  statements  are true  and statement II is not correct explanation of statement I cv) Statement I is true and statement II is false d)  Statement I is false and  Statement II is true

A.  Both statements  are  true and statement II is correct explanation of statement I

B.  Both  statements  are true  and statement II is not correct explanation of statement I

C.  Statement I is true and statement II is false

D.  Statement I is false and  Statement II is true

If a is any vector then (axi)2+(axj)2+(axk)2=

A.  a2

B.  2a2

C.  3a2

D.  4a2

The vector area of the parallelogram whose diagonals are i+j-2k, 2i-j+2k is

A.  1/2(i+4j-3k)

B.  1/2(i-4j+3k)

C.  1/2(i+4j+3k)

D.  1/2(i-4j-3k)

The area of the parallelogram whose diagonals are i-3j+2k, -i+2j is

A.  4√29 sq.unit

B.  1/2 √21 sq.unit

C.  10√3 sq.unit

D.  1/2√270 sq.unit

(2i-3j+k).(i-j+2k)x(2i+j+k)=

A.  -12

B.  14

C.  10

D.  15

Let v- = 2i- + j- - k- and u- = i- + 3k- . If u is any unit vector then the maximum value of the scalar triple product [u- v- w-] is

A.  1

B.  √10 + √6

C.  √59

D.  √60

I: If a=3i-2j+k, b=2i-4j-3k, c=-i+2j+2k then a+b+c=4i-4j II: If a=i-j+2k, b=2i+3j+k, c=i-k, then magnitude of a+2b-3c is √78

A.  Only I is true

B.  Only II is true

C.  both I and II are true

D.  Neither I nor II are true

If a=i+2j-3k, b=2i+j+k, c=i+3j-2k then (axb)x(bxc)=

A.  5(2i+j+k)

B.  -5(2i+j+k)

C.  10(2i+j+k)

D.  -10(2i+j+k)

A: If a, b, c are vectors such that [a b c]=4 then [axb bxc cxa]=64 R: [axb bxc cxa]=[a b c]2

A.  Both A and R are true and R is the correct explanation of A

B.  Both A and R are true but R is not correct explanation of A

C.  A is true R false

D.  A is false but R is true

If a,b,c are three non-collinear points then r=(1-p-q)a+pb+qc represents

A.  line

B.  plane

C.  plane passing through origin

D.  sphere

If a=i+j-2k , b=-i+2j+k, c=i-2j+2k then a unit vector parallel to a+b+c=

A.  2i+j+k/√6

B.  i+j+k/√3

C.  i-2j+k/√6

D.  i-j+k/√3

The vectors (1, 2, 3), (4, 5, 6), (6, 7, 8) are

A.  Linearly dependent

B.  linearly independent

C.  collinear

D.  none

The relation between the vectors a+3b+4c, a-2b+3c, a+5b-2c, 6a+14b+4c is

A.  1(a+3b+4c)+2(a-2b+3c)+2(a+5b-2c)-1(6a+14b+4c)=0

B.  1(a+3b+4c)+2(a-2b+3c)+3(a+5b-2c)-2(6a+14b+4c)=0

C.  1(a+3b+4c)+2(a-2b+3c)+3(a+5b-2c)-1(6a+14b+4c)=0

D.  none

The centre of the sphere (r-3i+3j+5k).(r+i-j+3k)=0 is

A.  (4, -6, 8)

B.  (2, -3, 4)

C.  (2, 2, 2)

D.  (1, 1, 1)