Properties of Vectors MCQ Quiz in বাংলা - Objective Question with Answer for Properties of Vectors - বিনামূল্যে ডাউনলোড করুন [PDF]

Last updated on Apr 22, 2025

পাওয়া Properties of Vectors उत्तरे आणि तपशीलवार उपायांसह एकाधिक निवड प्रश्न (MCQ क्विझ). এই বিনামূল্যে ডাউনলোড করুন Properties of Vectors MCQ কুইজ পিডিএফ এবং আপনার আসন্ন পরীক্ষার জন্য প্রস্তুত করুন যেমন ব্যাঙ্কিং, এসএসসি, রেলওয়ে, ইউপিএসসি, রাজ্য পিএসসি।

Latest Properties of Vectors MCQ Objective Questions

Top Properties of Vectors MCQ Objective Questions

Properties of Vectors Question 1:

If the magnitude of the sum of two non-zero vectors is equal to the magnitude of their difference, then which one of the following is correct?

  1. The vectors are parallel
  2. The vectors are perpendicular
  3. The vectors are anti-parallel
  4. The vectors must be unit vectors

Answer (Detailed Solution Below)

Option 2 : The vectors are perpendicular

Properties of Vectors Question 1 Detailed Solution

Concept:

Let u be a vector then |u|2= u u

Calculation:

Let u and v be two non-zero vectors.

Given: The magnitude of the sum of two non-zero vectors is equal to the magnitude of their difference.

|u+ v|= |u v|

|u+ v|2= |u v|2

As we know that, if u be a vector then |u|2= u u

(u+ v)(u+ v)=(u v)(u v)

|u|2+|v|2+2 |u|×|v|×cosθ= |u|2+|v|22 |u|×|v|×cosθ

4× |u|×|v|×cosθ=0

u and v be two non-zero vectors ⇒ |u|×|v|0

⇒ cos θ = 0 ⇒ θ = 90°

Hence, u and v are perpendicular to each other.

Properties of Vectors Question 2:

If the points (-1, -1, 2), (2, k, 5) and (3, 3, 6) are collinear, then find the value of k.

  1. 4
  2. 3
  3. 2
  4. 1

Answer (Detailed Solution Below)

Option 3 : 2

Properties of Vectors Question 2 Detailed Solution

Concept:

Conditions of collinear vector:

  • Three points with position vectors a,bandc are collinear if and only if the vectors (ab) and (ac) are parallel. ⇔ (ab)=λ(ac)
  • If the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) be collinear then |x1y1z1x2y2z2x3y3z3|=0

Calculation:

Let the given points be A (−1, −1, 2), B (2, K, 5), C (3, 11, 6).

Then AB=(2+1)i+(k+1)j+(52)k=3i+(k+1)j+3k

And AC=(3+1)i+(3+1)j+(22)k=4i+4j+4k

Now, if A, B and C are collinear, then AB=λAC

3i+(k+1)j+3k=λ(4i+4j+4k)

Comparing the coefficient of vector i, we get

⇒ 3 = 4 λ

∴ λ = 3/4

Now, comparing the coefficient of vector j, we get

⇒ (k + 1) = 4 λ

⇒ k + 1 = 4 × (3/4)

⇒ k + 1 = 3

∴ k = 2

Alternate solution:

We know that, If the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) be collinear then |x1y1z1x2y2z2x3y3z3|=0

Given (-1, -1, 2), (2, k, 5) and (3, 3, 6) are collinear

∴ |1122k5336|=0

⇒ -1 (6k – 15) – (-1) (12 – 15) + 2 (6 – 3k) = 0

⇒ -6k + 15 – 3 + 12 – 6k = 0

⇒ 12k = 24

∴ k = 2

Properties of Vectors Question 3:

If a and b are two non - zero vectors and their dot product 0 then, They are - 

  1. orthogonal
  2. Parallel
  3. at a certain angle
  4. None of these

Answer (Detailed Solution Below)

Option 1 : orthogonal

Properties of Vectors Question 3 Detailed Solution

Dot product of two vectors =ab=|a||b| cos α

If a and b are non zero vectors than if Dot product is to be 0 then.

Cos α = 0

α = 90°

ab

Properties of Vectors Question 4:

The area of the parallelogram determined by the vectors î + 2ĵ +3k̂ and 3î - 2ĵ + k̂ is

  1. 8√3
  2. 4√3
  3. 16√3
  4. 2√3

Answer (Detailed Solution Below)

Option 1 : 8√3

Properties of Vectors Question 4 Detailed Solution

Concept:

Area of parallelogram determined by the the vectors a and b is |a × b|.

Explanation:

Given

a = î + 2ĵ +3k̂ and b = 3î - 2ĵ + k̂

So, 

a × b = |i^j^k^123321|

    = î(2 + 6) + ĵ(9 - 1) + k̂(-2 - 6) = 8î + 8ĵ - 8k̂

Hence area of the parallelogram 

= |a × b| = 82+82+(8)2 = 64+64+64  = 8√3

Option (1) is true.

Properties of Vectors Question 5:

The position vector of the point which divides the join of points 2a3b and a+b in the ratio 3 ∶ 1 is

  1. 3a2b2
  2. 7a8b4
  3. 3a4
  4. 5a4

Answer (Detailed Solution Below)

Option 4 : 5a4

Properties of Vectors Question 5 Detailed Solution

Concept:

The position vector of the point that divides the line joining position vectors

p and q in the ratio m:n is given by mq+npm+n.

Calculation:

Given position vectors are  2a3b  and  a+b .

∴ The position vector of the point which divides the line joining the above points in the ratio 3: 1 is,

(2a3b)+3(a+b)1+3

=  5a4

The position vector of the point which divides the join of points 2a3b and a+b in the ratio 3 ∶ 1 is 5a4.

The correct answer is option 4.

Properties of Vectors Question 6:

If three vectors 2î - ĵ + k̂, î + 2ĵ - 3k̂ and 3î + λĵ + 5k̂ are co-planar, then λ is:

  1. -1
  2. -2
  3. -3
  4. -4

Answer (Detailed Solution Below)

Option 4 : -4

Properties of Vectors Question 6 Detailed Solution

Concept:

For three vectors AB and C to be co-planar, the volume of the parallelepiped formed by them must be 0.​ i.e. [A B C] = 0.

 

Triple Scalar Product (Box Product): is defined as: [A B C]=A.(B×C)=|a1a2a3b1b2b3c1c2c3|.

 

Calculation:

Let the three vectors be A=2i^j^+k^B=i^+2j^3k^ and C=3i^+λj^+5k^. For the three vectors to be co-planar, their Box Product must be 0.

⇒ [A B C]=0

⇒ |21   11   233   λ   5|=0

⇒ 2[(2)(5) - (-3)(λ)] - (-1)[(5)(1) - (3)(-3)] + 1[(1)(λ) - (2)(3)] = 0

⇒ 2(10 + 3λ) + 1(5 + 9) + 1(λ - 6) = 0

⇒ 20 + 6λ + 14 + λ - 6 = 0

⇒ 7λ = -28

⇒ λ = -4.

 

Additional Information

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

  • Dot Product is defined as A.B=|A||B|cosθ.
  • Cross Product is defined as A×B=n|A||B|sinθ where n is the unit vector perpendicular to the plane containing A and B.

 

Volume of a parallelepiped, with vectors AB and C as its sides, is given by the box product of the three vectors.

  • Volume = [A B C].

 

For three vectors AB and C:

  • Triple Cross Product: is defined as: A×(B×C)=(A.C)B(A.B)C.

Properties of Vectors Question 7:

The position vectors of three consecutive vertices of a parallelogram are i + j + k, i + 3j + 5k and 7i + 9j + 11k the position vector of the fourth vertex is

  1. 6(i - j + k)
  2. 7(i + j + k)
  3. 2j - 4k
  4. 6i + 8j + 10k

Answer (Detailed Solution Below)

Option 2 : 7(i + j + k)

Properties of Vectors Question 7 Detailed Solution

Concept:

Diagonals of a parallelogram bisect each other
Calculation:

Given:

Let A(1,1,1), B(1,3,5), C(7,9,11) and D(x,y,z) be the vertices of a parallelogram

As we know,

Diagonals of a parallelogram bisect each other

∴ midpoint of AC = midpoint of BD

(1+72,1+92,1+112)=(1+x2,3+y2,5+z2)

Comparing both sides, we get

1 + 7 = 1 + x

x = 7

And, 1 + 9 = 3 + y

y = 7

And, 1 + 11 = 5 + z

z = 7

 Position vector of fourth vertex is 7(i + j + k)

Properties of Vectors Question 8:

Let |a|0,|b|0.

(a+b).(a+b)=|a|2+|b|2

Holds if and only if

  1. a and b are perpendicular
  2. a and b are parallel
  3. a and b are inclined at an angle of 45°
  4. a and b are anti-parallel

Answer (Detailed Solution Below)

Option 1 : a and b are perpendicular

Properties of Vectors Question 8 Detailed Solution

Concept:

  • a and b are two vectors perpendicular to each other
  • ∴ a.b=0
  • a and b are two vectors parallel to each other
  • ∴ a×b=0


Calculation:

Given:

(a+b).(a+b)=|a|2+|b|2

|a|2+|b|2+2a.b=|a|2+|b|2

2a.b=0

a.b=0

a and b are perpendicular.

Properties of Vectors Question 9:

Consider the following statements :

1. Dot product over vector addition is distributive

2. Cross product over vector addition is distributive

3. Cross product of vectors is associative

Which of the above statements is/are correct ?

  1. 1 only
  2. 2 only
  3. 1 and 2 only
  4. 1, 2 and 3

Answer (Detailed Solution Below)

Option 3 : 1 and 2 only

Properties of Vectors Question 9 Detailed Solution

Concept:

One algebraic property of real numbers is the distributive law. The distributive law for the real numbers says: "For all real numbers x, y, and z, x.(y+z)=x.y+x.z

The vector dot product is distributive over addition. In general: a.(b+c)=a.b+a.c  

The vector cross product is distributive over addition. In general: a×(b+c)=a×b+a×c

Associative property: (p × q) × r = p × (q × r) (where p, q, and r are any three natural/whole numbers)

Calculation:

Let

a=axi+ayj+azkb=bxi+byj+bzkc=cxi+cyj+czk

Statement I: Dot product over vector addition is distributive

We have to prove a.(b+c)=a.b+a.c

  • a.(b+c)=(ax̂i+aŷj+aẑk).[(bx̂i+bŷj+bẑk)+(cx̂i+cŷj+cẑk)]
  • a.(b+c)=(ax̂i+aŷj+aẑk).[(bx+cx)̂i+(by+cy)̂j+(bz+cz)̂k]
  • a.(b+c)= a(b+ cx) + a(by + cy) + a(b+ cz)
  • a.(b+c)= ab+ ax cx + aby + acy + ab+ acz................................... (1)
  • a.b+a.c=(ax̂i+aŷj+aẑk).(bx̂i+bŷj+bẑk)+(ax̂i+aŷj+aẑk).(cx̂i+cŷj+cẑk)
  • a.b+a.c=(ax.bx+ay.by+az.bz)+(ax.cx+ay.cy+az.cz)
  • a.b+a.c= ab+ ax cx + aby + acy + ab+ acz................................... (2) 
  • From equation (1) and (2)
  • ∴ a.(b+c)=a.b+a.c  

Statement II: Cross product over vector addition is distributive

We have to prove a×(b+c)=a×b+a×c

  • a×(b+c)=(ax̂i+aŷj+aẑk)×[(bx̂i+bŷj+bẑk)+(cx̂i+cŷj+cẑk)]
  • a×(b+c)=(ax̂i+aŷj+aẑk)×[(bx+cx)̂i+(by+cy)̂j+(bz+cz)̂k]
  • a×(b+c)=[ijkaxayazbx+cxby+cybz+cz]
  • a×(b+c)= ̂̂î [a(b+ cz) - a(b+ cy)] - ĵ [a(bz + cz) - a(bx + cx)] + k̂ [a(b+ cy) - a(bx + cx)] ............(3) 
  • a×b+a×c=(ax̂i+aŷj+aẑk)×(bx̂i+bŷj+bẑk)+(ax̂i+aŷj+aẑk)×(cx̂i+cŷj+cẑk)
  • a×b+a×c=[̂îĵkaxayazbxbybz]+[̂îĵkaxayazcxcycz]
  • (a×b+a×c)= ̂̂î [aybz - azby)] - ĵ (axbz - azbx)] + k̂ [aby - aybx] + î [ayc- azcy)] - ĵ (axcz - azcx)] + k̂ [acy - aycx]
  • (a×b+a×c)= î [a(b+ cz) - a(b+ cy)] - ĵ [a(bz + cz) - a(bx + cx)] + k̂ [a(b+ cy) - a(bx + cx)] ..............(4) 
  • ∴ a×(b+c)=a×b+a×c

Statement III: Cross product of vectors is associative

  • Consider two non-zero perpendicular vectors, a and b.
  • We have (a × a) × b = 0 × b = 0
  • However, a × b is perpendicular to a and is not the zero vector, so
  • a × (a × b) ≠ 0
  • (a × a) × b ≠ a × (a × b)
  • Cross product of vectors is not associative

∴ Only Statements I and II are correct.

Properties of Vectors Question 10:

For aandb which of the following properties hold?

1. If aandb are two vectors parallel to each other then a×b=0

2.  If aandb are two vectors perpendicular to each other then ab=0

3. A cross product is commutative a×b=b×a

Select the correct answer using code given below:

  1. 1 and 2 only
  2. 2 and 3 only
  3. 1 and 3 only
  4. 1,2 and 3

Answer (Detailed Solution Below)

Option 1 : 1 and 2 only

Properties of Vectors Question 10 Detailed Solution

Concept:

Properties of vectors:

  • If aandb are two vectors parallel to each other then a×b=0
  • If aandb are two vectors perpendicular to each other then ab=0
  • A cross product is not commutative a×b=b×a

Hence option 1 is the correct answer.

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