Properties of Vectors MCQ Quiz in मराठी - Objective Question with Answer for Properties of Vectors - मोफत PDF डाउनलोड करा

Concept:

Let  be a vector then

Calculation:

Let  be two non-zero vectors.

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

As we know that, if  be a vector then

∵  be two non-zero vectors ⇒ 

⇒ cos θ = 0 ⇒ θ = 90°

Hence,  are perpendicular to each other.

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 

Calculation:

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

Then 

And 

Now, if A, B and C are collinear, then 

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 

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

∴ 

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

⇒ -6k + 15 – 3 + 12 – 6k = 0

⇒ 12k = 24

∴ k = 2

Dot product of two vectors  cos α

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

Cos α = 0

α = 90°

Concept:

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

Explanation:

Given

 = î + 2ĵ +3k̂ and  = 3î - 2ĵ + k̂

So, 

 ×  = 

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

Hence area of the parallelogram 

= | × | =  =   = 8√3

Option (1) is true.

Concept:

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

 and  in the ratio m:n is given by .

Calculation:

Given position vectors are    and   .

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

= 

=  

The position vector of the point which divides the join of points  and  in the ratio 3 ∶ 1 is .

The correct answer is option 4.

Concept:

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

Triple Scalar Product (Box Product): is defined as: .

Calculation:

Let the three vectors be ,  and . For the three vectors to be co-planar, their Box Product must be 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  and  at an angle θ to each other:

• Dot Product is defined as .
• Cross Product is defined as  where  is the unit vector perpendicular to the plane containing  and .

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

• Volume = .

For three vectors ,  and :

• Triple Cross Product: is defined as: .

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

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)

Concept:

•  and  are two vectors perpendicular to each other
• ∴ 
•  and  are two vectors parallel to each other
• ∴ 

Calculation:

Given:

∴  and  are perpendicular.

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, 

The vector dot product is distributive over addition. In general:   

The vector cross product is distributive over addition. In general:

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

Calculation:

Let

Statement I: Dot product over vector addition is distributive

We have to prove 

•  a_x (b_x + c_x) + a_y (b_y + c_y) + a_z (b_z + c_z)
•  ax bx + ax cx + ay by + ay cy + az bz + az cz................................... (1)
•  ax bx + ax cx + ay by + ay cy + az bz + az cz................................... (2) 
• From equation (1) and (2)
• ∴   

Statement II: Cross product over vector addition is distributive

We have to prove 

• ​​
•  ̂̂î [ay (bz + cz) - az (by + cy)] - ĵ [ax (bz + cz) - az (bx + cx)] + k̂ [ax (by + cy) - ay (bx + cx)] ............(3) 
•  ̂̂î [aybz - azby)] - ĵ (axbz - a_zbx)] + k̂ [ax by - aybx] + î [aycz - azcy)] - ĵ (axcz - azcx)] + k̂ [ax cy - aycx]
•  î [ay (bz + cz) - az (by + cy)] - ĵ [ax (bz + cz) - az (bx + cx)] + k̂ [ax (by + cy) - ay (bx + cx)] ..............(4) 
• ∴ 

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.

Concept:

Properties of vectors:

• If  are two vectors parallel to each other then 
• If  are two vectors perpendicular to each other then 
• A cross product is not commutative 

Hence option 1 is the correct answer.