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

Let  are unit vectors. Suppose that  and the angle between B and  then

a) True

b) False

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

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