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
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
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
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\(\hat{i}\)+\(\hat{j}\)
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
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
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
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
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\(\hat{j}\)
|\(\vec{a}+\vec{b}\)|=\(\sqrt{10^2+9^2}=\sqrt{100+81}=\sqrt{181}\)
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
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
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\(\hat{k}\)
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
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.
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