Integer Answer Type Questions for JEE: Vector Algebra | Chapter-wise Tests for JEE Main & Advanced PDF Download

Q.1. A, B, C and D are any four points in the space. If   where Δ_ABC is the area of triangle ABC, then λ is equal to:

Ans. 4
Let P.V. of A, B, C and D be

Q.2. The vectors  have their initial points at (1, 1), the value of λ so that the vectors terminate on one straight line is

Ans.  9
Since initial point of  their terminal points will be  Now given all the vectors terminate on one straight line.  Hence  ⇒ λ₁ = 1 and λ = 9

Q.3. Find the volume of the paralleopiped whose edges are represented by

Ans. 7
The required volume of the parallelopiped is equal to the absolute value of

= 6 + 15 - 28 = -7
Neglecting the negative sign, we get the volume of the parallelopiped = 7.

Q.4. Let  be three non-coplanar vectors, and let  be the vectors defined by the relations
Then the value of the expression  is equal to

Ans. 3

Therefore, the given expression is equal to 1 + 0 + 1 + 0 + 1 + 0 = 3.

Q.5. The number of vectors of unit length perpendicular to vectors a Ξ (1, 1, 0) and  b Ξ (0, 1, 1) is 

Ans. 2
The vector of unit length perpendicular to the given vectors
Hence, there are two such vectors.

Q.6. Two given points P and Q in the rectangular cartesian coordinates lie on y = 2^{x + 2} such that  where î is a unit vector along the x - axis. Find the magnitude of

Ans. 10
Let P(x₁, y₁), Q(x₂, y₂) be the two points on y = 2x+2
 projection of  on x – axis, so x₁ = -1 ⇒ y₁ = 2
projection of  on x – axis, so x₂ = 2 ⇒ y₂ = 16
If  is a unit vector along y– axis, then

Q.7.  Given that  are the position vectors of points P and Q respectively. Find the equation for the plane passing through Q and perpendicular to the line PQ. What is the distance from the point (-1, 1, 1) to the plane?

Ans. 5

Equation of plane passing through Q and perpendicular to PQ is

    .....(1)

Hence, distance from  the plane (1) is

Q.9. Line L₁ is parallel to a vector  and passes through a point A (7,6,2) and the line L₂ is parallel to a vector  and passes through a point B (5, 3,4). Now a line L₃ parallel to a vector  intersects the lines L₁ and L₂ at points C and D respectively. Find.

Ans. 9

P.V. of C.

P.V.of D,
 and we know that

Hence, by comparing both  we get 3a + 2b – 2c = 2-2a + b+ 2c = 3
⇒ -4a + 3b + c  = - 2 ⇒ a = 2, b = 1, c = 3

Q.9.  are two non-collinear vectors then the points with position vectors  are collinear if   _______.

Ans. 0
Given points will be collinear if
(ℓ₂ - ℓ₁)+ (m₂ - m₁) b = λ [(ℓ₃ - ℓ₂)+(m₃ - m₂) ]
or, [ℓ₂ - ℓ₁ - λ (ℓ₃ - ℓ₂)] + [(m₂ - m₁) - λ (m₃ - m₂)]
As ,  are non-collinear,
 
⇒ (ℓ₂ - ℓ₁) (m₃ - m₂) = (m₂ - m₁) (ℓ₃ - ℓ₂)

Q.10. Let  are vertices of a triangle and its median through A is equally inclined to the positive directions of the axes. The value of 2λ - μ is equal to _______.

Ans. 2


But direction ratios of Ad should be

λ = 6, μ = 10
2λ - μ = 2.