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
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
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
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
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)