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

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.

2. Direction of λ\(\vec{a}\) and \(\vec{a}\) is same if λ is _______

a) imaginary

b) negative

c) positive

d) zero

View Answer

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

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 \(\sqrt{(1^2+1^2+1^2)} = \sqrt{3}\).

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

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

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\(\hat{k}\).

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

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

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.

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

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

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.

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

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.

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