Mathematics Questions and Answers – Addition of Vectors

«
»

This set of Mathematics 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\(\hat{i}\)+\(\hat{j}\)
|\(\vec{a}+\vec{b}\)|=\(\sqrt{4^2+1^2}=\sqrt{16+1}=\sqrt{17}\)
advertisement

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\(\hat{i}\)+\(\hat{j}\)
advertisement
advertisement

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\(\hat{k}\)
advertisement

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

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\(\hat{k}\)
advertisement

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 areas of Mathematics, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

advertisement
advertisement
Manish Bhojasia - Founder & CTO at Sanfoundry
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