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 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)

 If u, v and w are non–coplanar vectors and p, q are real numbers, then the equality [3u  pv  pw] – [pv̄  w̄  qū] – [2w̄  qv̄  qū] = 0  holds for

a) Exactly two values of (p, q)

b) More than two but not all values of (p, q)

c) All values of (p, q)

d) Exactly one value of (p, q)

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.