Vector or Cross Product MCQ Quiz - Objective Question with Answer for Vector or Cross Product - Download Free PDF
Last updated on Jun 14, 2025
Latest Vector or Cross Product MCQ Objective Questions
Vector or Cross Product Question 1:
The position vectors of three points A, B and C respectively, where
Answer (Detailed Solution Below)
Vector or Cross Product Question 1 Detailed Solution
Calculation:
Given,
The position vectors of points A, B, and C are
The expression to evaluate is:
First, substitute
Using the distributive property of the cross product:
Since
Substitute
Factor out
Since
∴ The final result is
Hence, the correct answer is option 1.
Vector or Cross Product Question 2:
What is 3α + 2β equal to if (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) is a null vector?
Answer (Detailed Solution Below)
Vector or Cross Product Question 2 Detailed Solution
Explanation:
Given:
⇒ (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) = 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 =
∴ Option (1) is correct
Vector or Cross Product Question 3:
The unit vector perpendicular to each of the vectors
Answer (Detailed Solution Below)
Vector or Cross Product Question 3 Detailed Solution
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.
Vector or Cross Product Question 4:
If
Answer (Detailed Solution Below)
Vector or Cross Product Question 4 Detailed Solution
Formula Used:
Dot product of two vectors:
Magnitude of a vector:
Sine of double angle:
Relation between sine and cosine:
Calculation:
Given:
Vector 1:
Vector 2:
⇒
⇒
⇒
⇒
⇒
⇒
⇒
⇒
∴ The value of
Hence option 3 is correct
Vector or Cross Product Question 5:
A vector of magnitude
Answer (Detailed Solution Below)
Vector or Cross Product Question 5 Detailed Solution
Formula Used:
Unit vector along the internal bisector of two vectors
Resultant vector
Calculation:
Given:
Vector 1:
Vector 2:
Magnitude of resultant vector:
⇒
⇒
⇒ Unit vector along bisector:
⇒
Hence option 4 is correct
Top Vector or Cross Product MCQ Objective Questions
Find the value of
Answer (Detailed Solution Below)
Vector or Cross Product Question 6 Detailed Solution
Download Solution PDFConcept:
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°
If
Answer (Detailed Solution Below)
Vector or Cross Product Question 7 Detailed Solution
Download Solution PDFGiven:
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
∴ The correct option is 3
If
Answer (Detailed Solution Below)
Vector or Cross Product Question 8 Detailed Solution
Download Solution PDFConcept:
Let
Calculation:
Given:
As we know,
⇒ 6 = 3 × 4 × cos θ
⇒ cos θ =
∴ θ = 60°
As we know that, If
What is
Answer (Detailed Solution Below)
Vector or Cross Product Question 9 Detailed Solution
Download Solution PDFConcept:
Calculation:
Given
Additional Information
Properties of Scalar Product
In terms of orthogonal coordinates for mutually perpendicular vectors, it is seen that
Properties of Vector Product
What is the vector perpendicular to both the vectors î - ĵ and î ?
Answer (Detailed Solution Below)
Vector or Cross Product Question 10 Detailed Solution
Download Solution PDFConcept:
Let
Hence
Calculation:
Let vector
Therefore,
= (î × î) - (ĵ × î)
= 0 - (-k̂)
= k̂
If
Answer (Detailed Solution Below)
Vector or Cross Product Question 11 Detailed Solution
Download Solution PDFConcept:
For three vectors
- Triple Cross Product: is defined as:
Calculation:
Given that:
⇒
⇒
⇒
⇒
Find sin θ if theta is the angle between the vectors
Answer (Detailed Solution Below)
Vector or Cross Product Question 12 Detailed Solution
Download Solution PDFConcept:
Cross product of vectors:
-
- If
then
Given:
⇒
⇒
⇒
⇒
As we know that,
⇒
Hence, the correct option is 3.
A vector is perpendicular to both the vectors
Answer (Detailed Solution Below)
Vector or Cross Product Question 13 Detailed Solution
Download Solution PDFConcept:
Let
Hence
Calculation:
Let vector
Let
Therefore,
= (î + ĵ) × (ĵ + k̂)
= î × ĵ + î × k̂ + ĵ × ĵ + ĵ × k̂
= k̂ - ĵ + 0 + î
= î - ĵ + k̂
If
Answer (Detailed Solution Below)
Vector or Cross Product Question 14 Detailed Solution
Download Solution PDFConcept:
Let
Dot product of two vectors is given by:
Cross product of two vectors is given by:
Calculation:
Given:
To Find: Angle between
⇒ sin θ = cos θ (∵ |
⇒ tan θ = 1
∴ θ = 45°
What is
Answer (Detailed Solution Below)
Vector or Cross Product Question 15 Detailed Solution
Download Solution PDFConcept:
Properties of vectors:
If
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 2 is correct.