Vector Algebra MCQ Quiz - Objective Question with Answer for Vector Algebra - Download Free PDF

Concept:

Vector Operations and Identities:

• Dot Product: A scalar defined as 𝒂 · 𝒃 = |𝒂||𝒃|cosθ. It measures the projection of one vector on another.
• Cross Product: A vector defined as 𝒂 × 𝒃 = |𝒂||𝒃|sinθ 𝒏̂, perpendicular to the plane of 𝒂 and 𝒃.
• Vector Triple Product Identity: 𝒂 × (𝒃 × 𝒄) = (𝒂 · 𝒄)𝒃 − (𝒂 · 𝒃)𝒄
• Unit Vector: A vector of magnitude 1. If |𝒂| = 1, then 𝒂 is a unit vector.
• Regular Tetrahedron: A solid with 6 equal edges and 4 equilateral triangle faces. Angle between adjacent edges is 60°.

Calculation:

Given,

Statement A: 𝒂, 𝒃, 𝒄 form sides BC, CA, AB of ΔABC

⇒ 𝒂 + 𝒃 + 𝒄 = 0

⇒ Square both sides: (𝒂 + 𝒃 + 𝒄)² = 0

⇒ 𝒂² + 𝒃² + 𝒄² + 2(𝒂·𝒃 + 𝒃·𝒄 + 𝒄·𝒂) = 0

⇒ Let |𝒂| = |𝒃| = |𝒄| = 1

⇒ 3 + 2(𝒂·𝒃 + 𝒃·𝒄 + 𝒄·𝒂) = 0

⇒ 𝒂·𝒃 + 𝒃·𝒄 + 𝒄·𝒂 = −3/2

Statement B: 𝒂, 𝒃, 𝒄 are adjacent sides of regular tetrahedron

⇒ Angle between adjacent edges = 60°

⇒ 𝒂·𝒃 = 𝒃·𝒄 = 𝒄·𝒂 = cos 60° = 1/2

Statement C: 𝒂 × 𝒃 = 𝒄 ; 𝒃 × 𝒄 = 𝒂

⇒ Take LHS: 𝒂 × 𝒃 = 𝒄

⇒ Then 𝒃 × 𝒄 = 𝒂, and 𝒄 × 𝒂 = 𝒃 must also hold

⇒ Hence, 𝒂 × 𝒃 = 𝒃 × 𝒄 = 𝒄 × 𝒂

Statement D: 𝒂, 𝒃, 𝒄 are unit vectors & 𝒂 + 𝒃 + 𝒄 = 0

⇒ As before, square both sides:

⇒ 𝒂² + 𝒃² + 𝒄² + 2(𝒂·𝒃 + 𝒃·𝒄 + 𝒄·𝒂) = 0

⇒ 3 + 2(𝒂·𝒃 + 𝒃·𝒄 + 𝒄·𝒂) = 0

⇒ 𝒂·𝒃 + 𝒃·𝒄 + 𝒄·𝒂 = −3/2

∴ Correct matching is: I-(T), II-(P), III-(Q), IV-(R)

Concept:

If two points A and B have position vectors  and  respectively, then the vector .

For two vectors  and  at an angle θ to each other:

• Dot Product is defined as: .
• Resultant Vector is equal .
• Work: The work (W) done by a force () in moving (displacing) an object along a vector  is given by: W = .

Calculation:

Let's say that the forces acting on the particle are  = 2î - 5ĵ + 6k̂ and  = -î + 2ĵ - k̂.

∴ The resulting force acting on the particle will be .

⇒  = (2î - 5ĵ + 6k̂) + (-î + 2ĵ - k̂)

⇒  = î - 3ĵ + 5k̂.

Since the particle is moved from the point 4î - 3ĵ - 2k̂ to the point 6î + ĵ - 3k̂, the displacement vector  will be:

= (6î + ĵ - 3k̂) - (4î - 3ĵ - 2k̂)

⇒ ​ = 2î + 4ĵ - k̂.

And finally, the work done W will be:

W =  = (î - 3ĵ + 5k̂).(2î + 4ĵ - k̂)

⇒ W = (1)(2) + (-3)(4) + (5)(-1)

⇒ W = 2 - 12 - 5 =

∴ -15 units.

Concept:

If  are two vectors parallel to each other then  or 

Calculation:

Given:

  and  are parallel vectors,

Therefore, 

Equating the coefficient of 

⇒ 1 = 3λ, ∴ λ = 1/3            

⇒ -a = -6λ 

⇒ 5 = bλ                 .... (1)

Put the value of λ in equation (1), we get

5 = b × (1/3)

So, b = 15

Concept:

Magnitude of vector  then magnitude of vector is given by 

Calculation:

Given: Let where 

⇒ 

As we know that, if  then 

⇒ |

⇒  

Hence, option 1 is correct.

Concept:

Equal Vectors
Two or more vectors are said to be equal when their magnitude is equal and also their direction is the same.

Calculation:

Given:  and  

∴ a₃ = 1

Concept:

Conditions of collinear vector:

• Three points with position vectors  are collinear if and only if the vectors  and  are parallel. ⇔ 
• If the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) be collinear then 

Solution:

We know that, If the points (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3) be collinear then 

Given  , ,  are collinear

∴ 

⇒ 1 (-20 + 14) – (2) (-5λ + 21) + 3 (-2λ + 12) = 0

⇒ -6 + 10λ – 42 - 6λ + 36  = 0

⇒ 4λ = 12

∴ λ = 3

Concept:

Let  then magnitude of the vector of a = 

Calculation:

Let  =  p(2î - ĵ + 2k̂)

Given, 

⇒ 

⇒ 

⇒ 3p = 3

∴ p = 1

Concept:

Dot product of two vectors is defined as:

Cross/Vector product of two vectors is defined as:

where θ is the angle between 

Calculation:

To Find: Value of 

Here angle between them is 0°

Concept:

If , then 

Calculation:

Given A =  and B = 

 = 

 = 

Now 

 = 9√2

Concept:

Three or more points are collinear, if slope of any two pairs of points is same.

Here, 

Let, A = (5, -2), B = (8, -3), C = (a, -12)

Now, slope of AB = Slope of BC = Slope of AC ....(∵ points are collinear)

⇒ a - 8= 27

⇒ a = 27 + 8 = 35

Hence, option (4) is correct.

Concept:

For two vectors  to be collinear,​  where λ is a scalar.

Calculation:

Given that, the vectors  &  are collinear.

Since two vectors  are collinear then  where λ is a scalar.

⇒ 

⇒ 

⇒ λp = 4,  λq = 1 and -2λ = -3

⇒  λ = 3/2 

So, by substituting λ = 3/2 in  λp = 4 and λq = 1, we get

⇒ (3/2)p = 4 and (3/2)q = 1

⇒ p = 8/3 and q  = 2/3

∴  is the correct answer.

Concept:

If  then

Calculation:

Given:  and

Concept:

Let the angle between  and is 

Calculations:

consider, the angle between  and is 

Given, 

⇒

⇒

Squaring on both side, we get

⇒

⇒

⇒

⇒

⇒

⇒

⇒  = π / 3

Hence, If  and , then the angle between  and is π / 3

Concept:

If  are two vectors parallel to each other then  or 

Calculation:

Given:

  and  are parallel vectors,

Therefore, 

Equating the coefficient of 

⇒ 1 = 3λ, ∴ λ = 1/3            

⇒ -a = -6λ 

⇒ 5 = bλ                 .... (1)

Put the value of λ in equation (1), we get

5 = b × (1/3)

So, b = 15

Calculation:

  and c = î - ĵ - k̂

Given:  vector  in the plane of  and  

Therefore, 

⇒ 

= (1 + λ)î + (1 - λ)ĵ  + (1 + λ)k̂                .... (1)

Projection of  on 

⇒  

⇒ 

⇒ -(1 - λ) = 1

∴ λ = 2

Now, put the value of λ in equation (1), we get 

 = 3î - ĵ + 3k̂