# Class 12 Maths MCQ – Addition of Vectors

This set of Class 12 Maths Chapter 10 Multiple Choice Questions & Answers (MCQs) focuses on “Addition of Vectors”.

1. If $$\vec{a}$$=$$\hat{i}$$+4$$\hat{j}$$ and $$\vec{b}$$=3$$\hat{i}$$-3$$\hat{j}$$. Find the magnitude of $$\vec{a}+\vec{b}$$.
a) $$\sqrt{6}$$
b) $$\sqrt{11}$$
c) $$\sqrt{5}$$
d) $$\sqrt{17}$$
View Answer

Answer: d
Explanation: Given that, $$\vec{a}$$=$$\hat{i}$$+4$$\hat{j}$$ and $$\vec{b}$$=3$$\hat{i}$$-3$$\hat{j}$$
∴$$\vec{a}+\vec{b}$$=(1+3) $$\hat{i}$$+(4-3) $$\hat{j}$$
=4$$\hat{i}$$+$$\hat{j}$$
|$$\vec{a}+\vec{b}$$|=$$\sqrt{4^2+1^2}=\sqrt{16+1}=\sqrt{17}$$

2. Find the sum of the vectors $$\vec{a}$$=6$$\hat{i}$$-3$$\hat{j}$$ and $$\vec{b}$$=5$$\hat{i}$$+4$$\hat{j}$$.
a) 11$$\hat{i}$$+$$\hat{j}$$
b) 11$$\hat{i}$$–$$\hat{j}$$
c) -11$$\hat{i}$$+$$\hat{j}$$
d) $$\hat{i}$$+$$\hat{j}$$
View Answer

Answer: a
Explanation: Given that, $$\vec{a}$$=6$$\hat{i}$$-3$$\hat{j}$$ and $$\vec{b}$$=5$$\hat{i}$$+4$$\hat{j}$$
The sum of the vectors is given by $$\vec{a}+\vec{b}$$.
∴$$\vec{a}+\vec{b}$$=(6$$\hat{i}$$-3$$\hat{i}$$)+(5$$\hat{i}$$+4$$\hat{j}$$)
=(6+5) $$\hat{i}$$+(-3+4)$$\hat{j}$$
=11$$\hat{i}$$+$$\hat{j}$$

3. Find vector $$\vec{c}$$, if $$\vec{a}$$–$$\vec{b}$$+$$\vec{c}$$=6$$\hat{i}$$+8$$\hat{j}$$ where $$\vec{a}$$=7$$\hat{i}$$+2$$\hat{j}$$ and $$\vec{b}$$=4$$\hat{i}$$-5$$\hat{j}$$.
a) -3$$\hat{i}$$+$$\hat{j}$$
b) 3$$\hat{i}$$+$$\hat{j}$$
c) 3$$\hat{i}$$–$$\hat{j}$$
d) -3$$\hat{i}$$–$$\hat{j}$$
View Answer

Answer: b
Explanation: Given that, $$\vec{a}$$–$$\vec{b}$$+$$\vec{c}$$=6$$\hat{i}$$+8$$\hat{j}$$ -(1)
It is also given that, $$\vec{a}$$=7$$\hat{i}$$+2$$\hat{j}$$ and $$\vec{b}$$=4$$\hat{i}$$-5$$\hat{j}$$
Substituting the values of $$\vec{a}$$ and $$\vec{b}$$ in equation (1), we get
$$\vec{a}$$–$$\vec{b}$$+$$\vec{c}$$=6$$\hat{i}$$+8$$\hat{j}$$
(7$$\hat{i}$$+2$$\hat{j}$$)-(4$$\hat{i}$$-5$$\hat{j}$$)+$$\vec{c}$$=6$$\hat{i}$$+8$$\hat{j}$$
∴$$\vec{c}$$=(6$$\hat{i}$$+8$$\hat{j}$$)-(7$$\hat{i}$$+2$$\hat{j}$$)+(4$$\hat{i}$$-5$$\hat{j}$$)
=(6-7+4) $$\hat{i}$$+(8-2-5) $$\hat{j}$$
=3$$\hat{i}$$+$$\hat{j}$$
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4. Find the unit vector in the direction of the sum of the vectors, $$\vec{a}$$=2$$\hat{i}$$+7$$\hat{j}$$ and $$\vec{b}$$=$$\hat{i}$$-9$$\hat{j}$$.
a) $$\frac{3}{\sqrt{11}} \hat{i}-\frac{2}{\sqrt{11}} \hat{j}$$
b) $$\frac{2}{\sqrt{13}} \hat{i}-\frac{3}{\sqrt{13}} \hat{j}$$
c) –$$\frac{3}{\sqrt{11}} \hat{i}+\frac{2}{\sqrt{13}} \hat{j}$$
d) $$\frac{3}{\sqrt{13}} \hat{i}-\frac{2}{\sqrt{13}} \hat{j}$$
View Answer

Answer: d
Explanation: Given that, $$\vec{a}$$=2$$\hat{i}$$+7$$\hat{j}$$ and $$\vec{b}$$=$$\hat{i}$$-9$$\hat{j}$$
The sum of the two vectors will be
$$\vec{a}+\vec{b}$$=(2$$\hat{i}$$+7$$\hat{j}$$)+($$\hat{i}$$-9$$\hat{j}$$)
=(2+1) $$\hat{i}$$+(7-9)$$\hat{j}$$
=3$$\hat{i}$$-2$$\hat{j}$$
The unit vector in the direction of the sum of the vectors is
$$\frac{1}{|\vec{a}+\vec{b}|} (\vec{a}+\vec{b})=\frac{3\hat{i}-2\hat{j}}{\sqrt{3^2+(-2)^2}}=\frac{3\hat{i}-2\hat{j}}{\sqrt{13}}=\frac{3}{1\sqrt{3}} \hat{i}-\frac{2}{\sqrt{13}}\hat{j}$$

5. If $$\vec{a}$$=3$$\hat{i}$$+2$$\hat{j}$$+2$$\hat{k}$$, $$\vec{b}$$=2$$\hat{i}$$-8$$\hat{j}$$+$$\hat{k}$$, find $$\vec{a}+\vec{b}$$.
a) 5$$\hat{i}$$+$$\hat{j}$$+3$$\hat{k}$$
b) 5$$\hat{i}$$-6$$\hat{j}$$+3$$\hat{k}$$
c) 5$$\hat{i}$$-6$$\hat{j}$$-3$$\hat{k}$$
d) 5$$\hat{i}$$+6$$\hat{j}$$+3$$\hat{k}$$
View Answer

Answer: b
Explanation: It is given that, $$\vec{a}$$=3$$\hat{i}$$+2$$\hat{j}$$+2$$\hat{k}$$, $$\vec{b}$$=2$$\hat{i}$$-8$$\hat{j}$$+$$\hat{k}$$
To find: $$\vec{a}+\vec{b}$$
∴$$\vec{a}+\vec{b}$$=(3$$\hat{i}$$+2$$\hat{j}$$+2$$\hat{k}$$)+(2$$\hat{i}$$-8$$\hat{j}$$+$$\hat{k}$$)
=(3+2) $$\hat{i}$$+(2-8) $$\hat{j}$$+(2+1)$$\hat{k}$$
=5$$\hat{i}$$-6$$\hat{j}$$+3$$\hat{k}$$

6. Find the value of $$\vec{a}+\vec{b}$$+$$\vec{c}$$, if $$\vec{a}$$=4$$\hat{i}$$-4$$\hat{j}$$, $$\vec{b}$$=-3$$\hat{i}$$+2k, $$\vec{c}$$=7$$\hat{j}$$-8$$\hat{k}$$.
a) $$\hat{i}$$-3$$\hat{j}$$
b) $$\hat{i}$$+3$$\hat{j}$$-6$$\hat{k}$$
c) $$\hat{i}$$+$$\hat{j}$$+6$$\hat{k}$$
d) $$\hat{i}$$+6$$\hat{k}$$
View Answer

Answer: b
Explanation: Given that, $$\vec{a}$$=4$$\hat{i}$$-4$$\hat{j}$$, $$\vec{b}$$=-3$$\hat{i}$$+2k, $$\vec{c}$$=7$$\hat{j}$$-8$$\hat{k}$$
To find: $$\vec{a}+\vec{b}$$+$$\vec{c}$$
∴$$\vec{a}+\vec{b}$$+$$\vec{c}$$=(4$$\hat{i}$$-4$$\hat{j}$$) +(-3$$\hat{i}$$+2k) +(7$$\hat{j}$$-8$$\hat{k}$$)
=(4-3) $$\hat{i}$$+(-4+7) $$\hat{j}$$+(2-8)$$\hat{k}$$
=$$\hat{i}$$+3$$\hat{j}$$-6$$\hat{k}$$

7. Find the magnitude of $$\vec{a}+\vec{b}$$, if $$\vec{a}$$=4$$\hat{i}$$+9$$\hat{j}$$ and $$\vec{b}$$=6$$\hat{i}$$.
a) $$\sqrt{181}$$
b) $$\sqrt{81}$$
c) $$\sqrt{11}$$
d) $$\sqrt{60}$$
View Answer

Answer: a
Explanation: Given that, $$\vec{a}$$=4$$\hat{i}$$+9$$\hat{j}$$ and $$\vec{b}$$=6$$\hat{i}$$
∴$$\vec{a}+\vec{b}$$=(4+6) $$\hat{i}$$+9$$\hat{j}$$
=10$$\hat{i}$$+9$$\hat{j}$$
|$$\vec{a}+\vec{b}$$|=$$\sqrt{10^2+9^2}=\sqrt{100+81}=\sqrt{181}$$
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8. Find vector $$\vec{b}$$, if $$\vec{a}+\vec{b}$$+$$\vec{c}$$=8$$\hat{i}$$+2$$\hat{j}$$ where $$\vec{a}$$=$$\hat{i}$$-6$$\hat{j}$$ and $$\vec{c}$$=3$$\hat{i}$$+7$$\hat{j}$$.
a) 4$$\hat{i}$$+4$$\hat{j}$$
b) $$\hat{i}$$+4$$\hat{j}$$
c) 4$$\hat{i}$$–$$\hat{j}$$
d) 4$$\hat{i}$$+$$\hat{j}$$
View Answer

Answer: d
Explanation: Given that, $$\vec{a}+\vec{b}$$+$$\vec{c}$$=8$$\hat{i}$$+2$$\hat{j}$$ -(1)
Given: $$\vec{a}$$=$$\hat{i}$$-6$$\hat{j}$$ and $$\vec{c}$$=3$$\hat{i}$$+7$$\hat{j}$$
Substituting the values of $$\vec{a}$$ and $$\vec{b}$$ in equation (1), we get
$$\vec{a}+\vec{b}$$+$$\vec{c}$$=8$$\hat{i}$$+2$$\hat{j}$$
($$\hat{i}$$-6$$\hat{j}$$)+$$\vec{b}$$+(3$$\hat{i}$$+7$$\hat{j}$$)=8$$\hat{i}$$+2$$\hat{j}$$
∴$$\vec{c}$$=(8$$\hat{i}$$+2$$\hat{j}$$)-($$\hat{i}$$-6$$\hat{j}$$)-(3$$\hat{i}$$+7$$\hat{j}$$)
=(8-1-3) $$\hat{i}$$+(2+6-7) $$\hat{j}$$
=4$$\hat{i}$$+$$\hat{j}$$

9. If $$\vec{a}$$=9$$\hat{i}$$-2$$\hat{j}$$+7$$\hat{k}$$, $$\vec{b}$$=5$$\hat{i}$$+$$\hat{j}$$-3$$\hat{k}$$, find $$\vec{a}+\vec{b}$$.
a) $$\hat{i}$$–$$\hat{j}$$+4$$\hat{k}$$
b) 14$$\hat{i}$$–$$\hat{j}$$+4$$\hat{k}$$
c) 14$$\hat{i}$$-3$$\hat{j}$$+4$$\hat{k}$$
d) 14$$\hat{i}$$–$$\hat{j}$$+9$$\hat{k}$$
View Answer

Answer: b
Explanation: Given that, $$\vec{a}$$=9$$\hat{i}$$-2$$\hat{j}$$+7$$\hat{k}$$, $$\vec{b}$$=5$$\hat{i}$$+$$\hat{j}$$-3$$\hat{k}$$
We have to find $$\vec{a}+\vec{b}$$
∴$$\vec{a}+\vec{b}$$=(9$$\hat{i}$$-2$$\hat{j}$$+7$$\hat{k}$$)+(5$$\hat{i}$$+$$\hat{j}$$-3$$\hat{k}$$)
=(9+5) $$\hat{i}$$+(-2+1) $$\hat{j}$$+(7-3)$$\hat{k}$$
=14$$\hat{i}$$–$$\hat{j}$$+4$$\hat{k}$$
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10. Find the sum of the vectors $$\vec{a}$$=8$$\hat{i}$$+5$$\hat{j}$$ and $$\vec{b}$$=-2$$\hat{i}$$+6$$\hat{j}$$
a) 6$$\hat{i}$$+$$\hat{j}$$
b) 6$$\hat{i}$$+11$$\hat{j}$$
c) 6$$\hat{i}$$-11$$\hat{j}$$
d) $$\hat{i}$$+11$$\hat{j}$$
View Answer

Answer: b
Explanation: Given that, $$\vec{a}$$=8$$\hat{i}$$+5$$\hat{j}$$ and $$\vec{b}$$=-2$$\hat{i}$$+6$$\hat{j}$$
∴The sum of the vectors will be
$$\vec{a}+\vec{b}$$=(8$$\hat{i}$$+5$$\hat{j}$$)+(-2$$\hat{i}$$+6$$\hat{j}$$)
=(8-2) $$\hat{i}$$+(5+6)$$\hat{j}$$
=6$$\hat{i}$$+11$$\hat{j}$$

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

To practice all chapters and topics of class 12 Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.

If you find a mistake in question / option / answer, kindly take a screenshot and email to [email protected]

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