Let u and v be unit vectors. If w is a vector such that , then prove that  and that equality holds if and only if  is perpendicular to

If  is the unit vector along the incident ray,  is a unit vector along the reflected ray and is a unit vector along the outward drawn normal to the plane mirror at the point of incidence. Find  in terms of  and

True/False

If  for some non zero vector X then  

a) True

b) False

The points with position vectors  and  are collinear for all real values of k.

a) True

b) False

True / False

For any three vectors a, b and c
 

a) True

b) False

If  

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

Let  and c be two vectors perpendicular to each other in the XY–plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by

Let  and  where O, A and B are non-collinear points. Let p denote the area of the quadrilateral OABC and let q denote the area of the quadrilateral with OA and OC as adjacent sides. If p = kq then k = .  .  .  .  .

Let the vectors represent the edges of a regular hexagon

Statement 1 -  because

Statement 2 -

a) Statement 1 and 2 are true and Statement 2 is a correct explanation of statement 1.

b) Statement 1 and 2 are true and Statement 2 is not a correct explanation of statement 1.

c) Statement 1 is true. Statement 2 is false.

d) Statement 1 is false. Statement 2 is true.

The vector a = αî + 2ĵ + βk̂ lies in the plane of the vectors b = î + ĵ and c = ĵ + k̂ and bisects the angle between b and c. Then, which one of the following gives possible values of α and β

a) α = 1 and β = 1

b) α = 2 and β = 2

c) α = 1 and β = 2

d) α = 2 and β = 1

The non–zero vectors a, b and c are related by a = 8b and c = 7b. Then the angle between a and c is

a) π

b) 0

c)  

d)

If û and v̂ are unit vectors and θ is the acute angle between them, then 2û х 3v̂ is a unit vector for

a) Exactly two values of θ

b) More than two values of θ

c) No value of θ

d) Exactly one value of θ

Let a = î + ĵ + k̂, b = î – ĵ + 2k̂ and c = x î +(x − 2) ĵ  k̂. If the vector c lies in the plane of a and b, then x equals

a) 0

b) 1

c) 4

d) 2

Any three vectors are such that a . b ≠ 0,  b . c ≠ 0 then a and b are

a) inclined at an angle of  between them

b) perpendicular

c) parallel

d) inclined at an angle of  between them

The value of a, for which the points A, B, C with position vectors 2î – ĵ + k̂, î  3ĵ  5k̂ and a î  3ĵ + k̂ respectively are vertices of a right angle with C =  , are

a) 2 and −1

b) −2 and 1

c) 2 and −1

d) 2 and 1

Let C be the midpoint of AB and P is any point outside AB, then

a)  +  +  =

b)  +  +2  =

c)  +   =   

d)  +  = 2 

 The distance between the line r = 2î − 2ĵ + 3k̂ + λ (î – ĵ + 4k̂) and the plane r. (î + 5ĵ + k̂) = 5 is

a)

b)

c)

d)

For any vector a the value of (a x î)² + (a x ĵ)² + (a x k̂)² is equal to

a) 4a²

b) 2a²

c) a²

d) 3a²

Let a = î − k̂, b = xî + ĵ + (1 – x)k̂ and c = yî + xĵ + (1 + x – y)k̂. Then [a, b, c] depends on

a) Neither x nor y

b) Both x and y

c) Only x

d) Only y

Let a, b and c be distinct non–negative numbers. If the vectors aî + aĵ + ck̂, î + k̂ and cî + cĵ + bk̂ lie in a plane, then c is

a) harmonic mean of a and b

b) equal to zero

c) arithmetic mean of a and b

d) geometric mean of a and b

Let a, b and c be three non-zero vectors such that no two of these are collinear. If the vector a + 2b is collinear with c  and b + 3c is collinear with a  (λ being some non–zero scalar), then a + 2b + 6c equals

a) λa

b) λb

c) λc

d) 0

A particle is acted upon by constant forces 4î + ĵ – 3k̂ and 3î + ĵ − k̂ which displaces it from the point î + 2ĵ + 3k̂ to the point 5î + 4ĵ + k̂. The work done in standard units by the forces is given by

a) 40 units

b) 30 units

c) 25 units

d) 15 units

If , b and c are non–coplanar vectors and λ be a real number then the vectors a + 2b + 3c, λb + 4c and (2λ – 1) c are non-coplanar for

a) All values of λ

b) All except one value of λ

c) All except two values of λ

d) No value of λ