If u, v, w be such that |u| = 1, |v|= 2, |w| = 3. If the projection v along u is equal to that of w along u, and v, w are perpendicular to each other, then |u – v + w| equals

a) 2

b)

c)

d) 14

Let a, b and c be non–zero vectors such that (a x b) x c =  |b||c| a. If θ be the acute angle between b and c then sinθ equals

a)

b)

c)

d)

If a, b and c are three vectors such that a + b + c = , |a| = 1, |b| = 2, |c| = 3 then a.b + b.c + c.a is equal to

a) 0

b) −7

c) 7

d) 1

If u, v and w are three non–coplanar vectors then [u + v − w] . [(u – v)] x (v – w)] equals

a) 0

b) u . (v x w)

c) u . (w x v)

d) 3u . (v x w)

Consider points A, B, C and D with position vectors 7î − 4ĵ + 7k̂, î – 6ĵ + 10k̂, −î − 3ĵ + 4k̂ and 5î – ĵ + 5k̂ respectively. Then ABCD is a

a) Square

b) Rhombus

c) Rectangle

d) None of these

The vectors  = 3î + 4k̂ and  = 5î − 2ĵ + 4k̂ are sides of a triangle ABC. The length of median through A is

a)

b)

c)

d)

If   = 0 and vectors (1, a, a²), (1, b, b²) and (1, c, c²) are non-coplanar then the product abc equals

a) 2

b) −1

c) 1

d) 0

Let u = î + ĵ, v = î – ĵ and w = î + 2ĵ + 3k̂. If n̂ is an unit vector such that u . n̂ = 0, v . n̂ = 0 then |w . n̂| is equal to

a) 0

b) 1

c) 2

d) 3

Given two vectors î – ĵ and î + 2ĵ the unit vector coplanar with the two vectors and perpendicular to the first is

a)  

b)

c) ±

d) None of these

The vector î + x ĵ + 3k̂ is rotated through an angle θ and doubled in magnitude, then it becomes 4î + (4x – 2)ĵ + 2k̂. The values of x are

a)

b)

c)

d)

If the vectors a, b and c form the sides BC, CA and AB respectively of a ΔABC, then

a) a . b = b . c  = c . b = 0

b) a x b = b x c = c x a

c) a . b = b . c  = c . a = 0

d) a x a + a x c + c x a = 0

If the vectors , a = xî + yĵ + zk̂ and b = ĵ are such that a, c and b form a right handed system, then c is

a) zî − xk̂

b) 0

c) yĵ

d) –zî + xk̂

If â, b̂, ĉ are unit vectors such that â + b̂ + ĉ =  then the value of â . b̂ + b̂ . ĉ + ĉ . â is

a) 1

b) 3

c) −  

d) None of these

A parallelogram is constructed on the vectors a = 3α – β, b = α + 3β. If |α| = |β| = 2 and the angle between α and β is  then the length of a diagonal of the parallelogram is

a) 4

b) 4

c) 4

d) None of these

The value of c, so that for all real x the vectors cxî − 6ĵ + 3k̂ and xî + 2ĵ + 2cxk̂ make an obtuse angle are

a) c < 0

b) 0 < c <

c)  < c < 0

d) c > 0

If p =  , q =  , r =   where a, b, c are three non-coplaner vectors then the value of the expression (a + b + c) . (p + q + r) is

a) 3

b) 2

c) 1

d) 0

Let , ,  are three mutually perpendicular vectors of same magnitude. If  satisfies the equation  x {( – ) x } +  x {( – ) x } +  x {( –) x } =  then  is given by

a)  ( +  − 2)

b)  ( +  + )

c)  (2 +  − )

d)  ( +  − 2)

If ­a­ = ĵ − k̂ and c = î − ĵ − k̂, then the vector b satisfying a x b + c = 0 and a . b = 3 is

a) – î + ĵ – 2k̂

b) 2î – ĵ +2k̂

c) î − ĵ – 2k̂

d) î + ĵ – 2k̂

The vectors a = î − ĵ + 2k̂, b = 2î + 4ĵ + k̂ and c = λî + ĵ + μk̂ are mutually orthogonal then (λ, μ) is equal to

a) (−3, 2)

b) (2, −3)

c) (−2, 3)

d) (3, −2)

The projections of a vector on the three coordinate axes are 6, −3, 2 respectively then the direction cosines of the vectors are

a) 6, −3, 2

b) ,  

c) ,  

d) ,  

If ABCD are four points in a space, prove that

The unit vector perpendicular to the plane determined by
 is.