Mathematics Questions and Answers – Multiplication of a Vector by a Scalar

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This set of Mathematics Exam Questions and Answers for Class 12 focuses on “Multiplication of a Vector by a Scalar”.

1. Multiplication of vector \(\vec{a}\) and scalar λ is denoted as ______
a) λ\(\vec{a}\)
b) \(\vec{a}\)
c) λ
d) 0
View Answer

Answer: a
Explanation: Multiplication of vector \(\vec{a}\) and scalar λ is denoted as λ\(\vec{a}\), as \(\vec{a}\) is the original vector. λ is the scalar which can have any integer value which is to be multiplied to the given vector\(\vec{a}\), whereas 0 can only be the answer if the scalar λ = 0.
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2. Direction of λ\(\vec{a}\) and \(\vec{a}\) is same if λ is _______
a) imaginary
b) negative
c) positive
d) zero
View Answer

Answer: c
Explanation: Direction of λ\(\vec{a}\) and \(\vec{a}\) is same if value λ is positive as it gives it a direction which is positive in nature. If the value of λ is negative then the direction of the result after multiplication becomes in opposite direction. Whereas the value of the product vector becomes zero if value of λ is 0.

3. Find magnitude \(\vec{a}\) =\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\).
a) \(\sqrt{3}\)
b) \(\sqrt{2}\)
c) 0
d) \(\sqrt{4}\)
View Answer

Answer: a
Explanation: Magnitude of vector is calculated by formula \(\sqrt{x^2+ y^2+ z^2}\).
Where x, y, z are the coefficients of \(\hat{i}\), \(\hat{j}\), \(\hat{k}\).
The magnitude of vector \(\vec{a}\) is calculated as \(\sqrt{(1^2+1^2+1^2)} = \sqrt{3}\).
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4. |λ| times the magnitude of vector \(\vec{a}\) is denoted as ______
a) |λ\(\vec{a}\)|
b) λ|\(\vec{a}\)|
c) |λ|\(\vec{a}\)
d) λ\(\vec{a}\)
View Answer

Answer: a
Explanation: |λ| times the magnitude of vector \(\vec{a}\) is denoted as |λ\(\vec{a}\)| = |λ||\(\vec{a}\)|
As we know that the magnitude of vector \(\vec{a}\) is denoted by |\(\vec{a}\)|, if we multiply the magnitude of vector \(\vec{a}\) with magnitude of λ we get |λ\(\vec{a}\)|.

5. If \(\vec{a}\) =\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\) and λ=5, what is value of λ\(\vec{a}\)?
a) \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\)
b) 5\(\hat{i}\) + 5\(\hat{j}\) + 5\(\hat{k}\)
c) \(\hat{i}\) + 5\(\hat{j}\) + 5\(\hat{k}\)
d) 10\(\hat{i}\) + 10\(\hat{j}\) + 10\(\hat{k}\)
View Answer

Answer: b
Explanation: Multiplication of vector \(\vec{a}\) =\(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\) by scalar value 5 results in 5\(\hat{i}\) + 5\(\hat{j}\) + 5\(\hat{k}\), as in these type of questions we multiply\(\hat{i}\), \(\hat{j,}\) \(\hat{k}\) with the constant given and the answer comes out to be 5\(\hat{i}\) + 5\(\hat{j}\) + 5\(\hat{k}\).
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6. If k is any scalar and \(\vec{a}\), \(\vec{b}\) be vectors then k (\(\vec{a}\)+ \(\vec{b}\))= ________
a) k\(\vec{a}\) + k\(\vec{b}\)
b) k\(\vec{a}\) + \(\vec{b}\)
c) \(\vec{a}\) + k\(\vec{b}\)
d) \(\vec{a}\) + \(\vec{b}\)
View Answer

Answer: a
Explanation: Multiplication of vector by scalar satisfies distributive property over addition and in k (\(\vec{a}\)+ \(\vec{b}\)) we multiply k with \(\vec{a}\), \(\vec{b}\) individually and hence the answer comes out to be k\(\vec{a}\) + k\(\vec{b}\).

7. Find values of x, y, z if vectors \(\vec{a}\)=x\(\hat{i}\) + 2\(\hat{j}\) + z\(\hat{k}\) and \(\vec{b}\)=2\(\hat{i}\) + y\(\hat{j}\) + \(\hat{k}\) are equal.
a) x=2, y=2, z=1
b) x=1, y=2, z=1
c) x=2, y=1, z=1
d) x=2, y=2, z=2
View Answer

Answer: a
Explanation: As both the vectors are equal hence, we can equate their constants and get the value of x, y and z. Now we equate the coefficients of \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) of both the equations and get the values x=2, y=2, z=1.
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8. \(\vec{a}\)=\(\hat{i}\) + 2\(\hat{j}\) and \(\vec{b}\)=2\(\hat{i}\) + \(\hat{j}\) , Is |\(\vec{a}\)| = |\(\vec{b}\)|?
a) Yes
b) No
View Answer

Answer: a
Explanation: As we know that magnitude of vector is calculated by formula \(\sqrt{x^2+ y^2}\).
Therefore, |\(\vec{a}\)| = \(\sqrt{12} + 22 = \sqrt{5}\) and \(|\vec{b}| = \sqrt{22} + 12 = \sqrt{5}\), they are equal.

9. What is direction of vector \(\vec{a}\) if it is multiplied with -λ?
a) Downwards
b) Upwards
c) Same
d) Opposite
View Answer

Answer: d
Explanation: If the vector is multiplied with –λ then its direction become opposite as the direction in which it was previous may be positive or negative. After it is multiplied with a negative value then its direction becomes exactly opposite to the previous direction.
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10. If k is any scalar and \(\vec{a}\), \(\vec{b}\) be vectors then k \(\vec{a}\) + m\(\vec{a}\) can also be written as ________
a) (k+m)\(\vec{a}\)
b) \(\vec{a}\) + m\(\vec{a}\)
c) k \(\vec{a}\) + \(\vec{a}\)
d) mk\(\vec{a}\)
View Answer

Answer: a
Explanation: It satisfies distribution property over addition, hence in k \(\vec{a}\) + m\(\vec{a}\) we can take the vector \(\vec{a}\)
common and the answer come out to be (k+m)\(\vec{a}\). Basically it’s a simplification method by which the vectors can be easily solved and further properties can be applied to them.

Sanfoundry Global Education & Learning Series – Mathematics – Class 12.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn | Youtube | Instagram | Facebook | Twitter