# 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

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

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) $$\sqrt{4}$$

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$$\vec{a}$$

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$$\hat{k}$$

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) $$\vec{a}$$ + $$\vec{b}$$

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

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

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

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$$\vec{a}$$

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.