Vector or Cross Product MCQ Quiz - Objective Question with Answer for Vector or Cross Product - Download Free PDF

Calculation:

Given,

The position vectors of points A, B, and C are , , and respectively, and .

The expression to evaluate is: .

First, substitute into the equation:

.

Using the distributive property of the cross product:

.

Since and , we are left with:

.

Substitute into the expression:

.

Factor out

.

Since , the expression becomes:

.

∴ The final result is .

Hence, the correct answer is option 1. 

Explanation:

Given:

⇒ (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) = 0

⇒  = 0

⇒ 

Comparing both sides, we get

6β  – 27α  = 0

⇒ 2β  = 9α 

2β - 27 = 0

⇒ β = 27/2

Also

⇒ 2α – 6 = 0 

α =3

Now, 

3α  + 2β =  =36

∴ Option (1) is correct

Concept:

Vector Perpendicular to Both Vectors:

• We are given two vectors: a + b and a - b, and we need to find a vector perpendicular to both of them.
• The method to find a vector perpendicular to both given vectors is by taking their cross product. The result will be a vector perpendicular to both.
• Once the cross product is found, we normalize the result (i.e., divide it by its magnitude) to find the unit vector perpendicular to both vectors.

Calculation:

Given vectors:

a = i + j + k

b = i + 2j + 3k

The vectors we need to take the cross product of are:

a + b = (i + j + k) + (i + 2j + 3k) = 2i + 3j + 4k

a - b = (i + j + k) - (i + 2j + 3k) = 0i - j - 2k

Now, we compute the cross product of (a + b) and (a - b):

(a + b) × (a - b) =

Expanding the determinant:

Result of cross product:

(a + b) × (a - b) = -2i + 4j - 2k

Now, let's find the magnitude of the resulting vector:

Magnitude = √((-2)² + 4² + (-2)²) = √(4 + 16 + 4) = √24 = 2√6

To find the unit vector, we divide the result by its magnitude:

Unit vector = (-2i + 4j - 2k) / 2√6

The unit vector is:

Unit vector = (-1/√6)i + (2/√6)j - (1/√6)k

Hence Option 4 is the correct answer. 

Formula Used:

Dot product of two vectors: , where is the angle between the vectors.

Magnitude of a vector:

Sine of double angle:

Relation between sine and cosine:

Calculation:

Given:

Vector 1: 

Vector 2: 

⇒

⇒

⇒

⇒

⇒

⇒

⇒

⇒

∴ The value of is

Hence option 3 is correct

Formula Used:

Unit vector along the internal bisector of two vectors and is given by:

Resultant vector where is the unit vector along the bisector and is the magnitude of the resultant vector.

Calculation:

Given:

Vector 1: 

Vector 2: 

Magnitude of resultant vector: 

⇒

⇒

⇒ Unit vector along bisector:

⇒

Hence option 4 is correct

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°

Given: 

Concept:

î × î  = ĵ × ĵ = k̂ × k̂  = 0 

î × ĵ = k̂ , ĵ × k̂ = î , k̂ × î = ĵ 

Calculation:

Let a = mî + nĵ +lk̂ 

According to the Question 

 = î  × (mî + nĵ +lk̂  × î) + ĵ ×  (mî + nĵ +lk̂  × ĵ) + k̂ × (mî + nĵ +lk̂  × k̂)

 =  î  × (-nk̂ + lĵ) + ĵ × (mk̂ -lî  ) + k̂ × (-mĵ + nî) 

 = nĵ  + lk̂ + mî +  lk̂ + mî + nĵ 

 = 2(mî + nĵ +lk̂ ) = 2

∴ The correct option is 3

Concept:

Let  are two vectors

  is the unit vector perpendicular to both 

Calculation:

Given: 

As we know, 

⇒ 6 = 3 × 4 × cos θ 

⇒ cos θ = 

∴ θ = 60° 

As we know that, If  are two vectors, then

      (∵ Magnitude of a unit vector is one)

 = 3 × 4 × sin 60° 

Concept:  

Calculation:

Given

Additional Information

Properties of Scalar Product

 (Scalar product is commutative)

 (Distributive of scalar product over addition)

In terms of orthogonal coordinates for mutually perpendicular vectors, it is seen that 

Properties of Vector Product

Concept:

Let  and  be the two vectors and the vector  perpendicular to both  and 

Hence  

Calculation:

Let vector  is perpendicular to both the vectors î - ĵ and î

Therefore,  = (î - ĵ) × î

= (î × î) -  (ĵ × î)

= 0 - (-k̂)

= k̂

Concept:

For three vectors ,  and :

• Triple Cross Product: is defined as: 

Calculation:

Given that: 

⇒ 

⇒ 

⇒ 

⇒ , which is the required answer.

Concept:

Cross product of vectors:

•  
• If  then 

Given: 

⇒ 

⇒ 

⇒ 

⇒

As we know that, 

⇒ 

Hence, the correct option is 3.

Concept:

Let  and  be the two vectors and the vector  perpendicular to both  and 

Hence  

Calculation:

Let vector  is perpendicular to both the vectors î + ĵ and ĵ + k̂

Let  = î + ĵ and  = ĵ + k̂

Therefore, 

= (î + ĵ) × (ĵ + k̂)

= î × ĵ + î × k̂ + ĵ × ĵ + ĵ × k̂ 

= k̂ - ĵ + 0 + î 

= î - ĵ + k̂  

Concept:

Let  and  be the two vectors,

Dot product of two vectors  is given by:  

Cross product of two vectors  is given by: , Where  is a unit vector

Calculation:

Given: 

To Find: Angle between 

⇒ sin θ = cos θ                       (∵ || = 1)

⇒ tan θ = 1

∴ θ = 45°