Test: Comprehension Based Questions: Vector Algebra and Three Dimensional Geometry - JEE MCQ

Vector in the direction of 
Vector in the direction of   
∴ Vector perpendicular to both L₁ and L₂

The shortest distance between L₁ and L₂ is

The plane passing through (–1, –2, –1) and having normal along   
– 1(x + 1) – 7(y + 2) + 5(z + 1) = 0
or x + 7y - 5z +10 = 0
∴ Distance of point (1, 1, 1) from the above plane is

The line of intersection of given plane is 3x -6y - 2z -15 = 0 = 2x + y - 2z-5
For z = 0 , we obtain x = 3 and y =-1
∴Line passes through (3, –1, 0).
Let a, b, c be the d’rs of line of intersection, then 3a -6b -2c = 0 and 2a + b - 2c = 0

Solving the above equations using cross product method, we get a :b :c = 14 : 2 : 15

∴ Eqn. of line is 

whose parametric form is x = 3 + 14t , y = 1 + 2t, z= 15t

∴ Statement-I is false (value of y is not matching).
Since dr’s of line intersection of given planes are 14, 2,15

∴ is parallel to this line.
∴ Statement 2 is true.

Statement-1 is true.

∴ Statement-2 is false.

The given planes are
P₁ : x – y + z = 1 ...(1)
P₂ : x + y – z = –1 ...(2)
P₃ : x – 3y + 3z = 2 ...(3)

Line L₁ is intersection of planes P₂ and P₃.
∴ L₁ is parallel to the vector

Line L₂ is intersection of P₃ and P₁

∴ L₂ is parallel to the vector

Line L₃ is intersection of P₁ and P₂

∴ L₃ is parallel to the vector

Clearly lines L₁, L₂ and L₃ are parallel to each other.

∴ Statement–1 is False

Also family of planes passing through the intersection of P₁ and P₂ is P₁ + λP₂ = 0 .If plane P₃ is represented by P₁ + λP₂ = 0 for some value of l then the three planes pass through the same point.
Here P₁ + λP₂ = 0

This will be identical to P₃ if

...(1)

 and taking 

∴There is no value of λ which satisfies eq (1).

∴ The three planes do not have a common point.
⇒ Statement 2 is true.

∴ (d) is the correct option.