Properties of Vectors MCQ Quiz - Objective Question with Answer for Properties of Vectors - Download Free PDF

Calculation:

Given,

The vector is:

The vector is:

Substituting :

Step 4: Now, , which gives:

∴ The correct ratio is AB : BC = 1 : 3 , 

Hence, the correct answer is Option 2. 

Calculation:

Given,

The vector

Statement I: is coplanar with and .

We use the vector triple product identity: .

This shows that is a linear combination of and , hence is coplanar with and .

Therefore, Statement I is correct.

Statement II: is perpendicular to .

To check this, compute the dot product . Using the vector triple product identity, we find:

,

which means is perpendicular to .

Therefore, Statement II is correct.

∴ Both Statement I and Statement II are correct.

Hence, the correct answer is option  3. 

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:

If  then

Calculation:

Given:  and

Calculation

From triangular law of vector addition, we get

⇒ 

But (Given)

⇒ 

⇒ 

Hence option 3 is correct

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:

Calculation:

Given:   = 3,  and 

We know that,

∴

Concept:

•  and  are two vectors parallel to each other ⇔  
• Cross product of parallel vectors are zero ⇔  
• A cross or vector product is not commutative ⇔ 

Calculation:

We have to find the value of 

We know that 

∴ Option 3 is correct.

Concept:

If vectors  are perpendicular then 

Calculation:

Given:  and  are perpendicular

Let  and 

We know that, If vectors  are perpendicular then 

⇒ -2 - 20 - λ = 0 

⇒ -22 - λ = 0 

∴ λ = -22

Concept:

Scaler triple product of the vectors:

The scaler triple product of the vectors  and  is given by:

Coplaner vectors:

Three vectors  and  are said to be coplaner if the scaler triple product

Solution:

Let the given vectors be  and .

It is given that the vectors are coplaner therefore the scaler triple product

Therefore,

Perform the coloumn operation C₁ – C₂ as follows:

Now perform another coloum operation C₂ – C₃ as follows:

Take (1 - a)(1 - b)(1 - c) common. Note that it is given that a,b,c ≠ 0 therefore this action is jstified.

Since a, b, c ≠ 0 therefore (1 - a)(1 - b)(1 - c) ≠ 0.Therefore the determinant has to be zero.

Simplify the above expression as follows:

Therefore, .

Concept:

• If a, b, c are coplanar then [a b c] = 0
• Three vectors are permuted in the same cyclic order, the value of the scalar triple product remains the same. ⇒  [a b c] = [b c a] = [c a b]

Calculation:

Here, a, b, c are non-coplanar

To find: 2[a b c] + [b a c] =?

⇒ 2[a b c] + [b a c]

= 2 [a, b, c] - [a, b, c]                (∵ [b a c] = -[a b c])

= [a, b, c] 

Hence, option (2) is correct.

Concept:

If vectors a and b are orthogonal, then 

Calculation: 

Here, 

Squaring both sides we get, 

⇒4= 0

⇒= 0

So, Vectors a and b are orthogonal

As we know, if  then 

So, only (1) is correct.

Hence, option (1) is correct.

Concept:

The cross product of two vectors is given by  and 

The scalar product of two vectors is given by

If  is a unit vector then 

Calculations:

Statement 1: The cross product of two unit vectors is always a unit vector.

Let  and  are two unit vectors.

i.e 

As we know that, the cross product of two vectors is given by  and 

⇒ 

The range of sin θ is [-1, 1]

So, it is not necessarily true that the cross product of two unit vectors is always a unit vector.

Hence, statement 1 is false.

Statement 2: The dot product of two unit vectors is always unity.

Let  and  are two unit vectors.

i.e 

As we know that, the scalar product of two vectors is given by  

⇒ 

The range of cos θ is [-1, 1].

So, it is not necessarily true that the dot product of two unit vectors is always a unit vector.

Hence, statement 2 is false.

Statement 3: The magnitude of sum of two unit vectors is always greater than the magnitude of their difference.

Let  

As we can see that, the vectors   and  are two unit vectors

⇒   and 

⇒ 

So, statement 3 is also false.

Hence, the correct option is 4.

Concept:

Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.

From above statement,

Only statement (1) is correct