Mathematics Questions and Answers – Product of Two Vectors-1

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This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Product of Two Vectors-1”.

1. Find the scalar product of the vectors \(\vec{a}=2\hat{i}+5\hat{j}\) and \(\vec{b}=6\hat{i}-7\hat{j}\).
a) -32
b) -23
c) 32
d) 23
View Answer

Answer: b
Explanation: If \(\vec{a} \,and \,\vec{b}\) are two vectors, where a1, a2 are the components of vector \(\vec{a} \,and \,b_1, \,b_2\) are the components of vector \(\vec{b}\), then the scalar product is given by
\(\vec{a}.\vec{b}=a_1 \,b_1+a_1 \,b_2\)
∴\((2\hat{i}+5\hat{j}).(6\hat{i}-7\hat{j})\)=2(6)+5(-7)=12-35=-23.
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2. Find the angle between the two vectors \(\vec{a}\) and \(\vec{b}\) with magnitude \(\sqrt{3}\) and \(\sqrt{2}\) respectively and \(\vec{a.} \,\vec{b}=3\sqrt{2}\).
a) \(cos^{-1}⁡\frac{1}{\sqrt{3}}\)
b) \(cos^{-1}⁡\sqrt{3}\)
c) \(cos^{-1}⁡\frac{3}{\sqrt{2}}\)
d) \(cos^{-1}⁡\frac{2}{\sqrt{3}}\)
View Answer

Answer: a
Explanation: Given that, \(|\vec{a}|=\sqrt{3} \,and \,|\vec{b}|=\sqrt{2}\)
Also, \(\vec{a.} \vec{b}=3\sqrt{2}\)
The angle between two vectors is given by
\(cos⁡θ=\frac{|\vec{a}|.|\vec{b}|}{\vec{a}.\vec{b}}\)
∴\(cos⁡θ=\frac{\sqrt{3}.\sqrt{2}}{3\sqrt{2}}=\frac{1}{\sqrt{3}}\)
∴\(θ=cos^{-1}⁡\frac{1}{\sqrt{3}}\).

3. Find the projection of vector \(\vec{a}=8\hat{i}-\hat{j}+6\hat{k}\) on vector \(\vec{b}= 4\hat{i}+3\hat{j}\).
a) \(\sqrt{\frac{29}{5}}\)
b) \(\frac{29}{\sqrt{5}}\)
c) \(\frac{\sqrt{29}}{5}\)
d) \(\frac{29}{5}\)
View Answer

Answer: d
Explanation: The projection of a vector \(\vec{a}\) on vector \(\vec{b}\) is given by
\(\frac{1}{|\vec{b}|} (\vec{a}.\vec{b})\)
\(|\vec{b}|=\sqrt{4^2+3^2}=\sqrt{16+9}\)=5
\(\vec{a}.\vec{b}\)=8(4)-1(3)+0=32-3=29
The projection of vector \(8\hat{i}-\hat{j}+6\hat{k}\) on vector \(4\hat{i}+3\hat{j}\) will be
\(\frac{1}{|\vec{b}|} (\vec{a}.\vec{b})=\frac{1}{5} (29)=\frac{29}{5}\)
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4. Find \(|\vec{a}+\vec{b}|\), if \(|\vec{a}|=3 \,and \,|\vec{b}|=4 \,and \,\vec{a}.\vec{b}=6\).
a) 34
b) \(\sqrt{37}\)
c) 13
d) \(\sqrt{23}\)
View Answer

Answer: b
Explanation: \(|\vec{a}+\vec{b}|^2=(\vec{a}+\vec{b}).(\vec{a}+\vec{b})\)
=\(\vec{a}.\vec{a}+\vec{a}.\vec{b}+\vec{b}.\vec{a}+\vec{b}.\vec{b}\)
=\(|\vec{a}|^2+2(\vec{a}.\vec{b})+|\vec{b}|^2\)
=(3)2+2(6)+(4)2
=9+12+16=37
∴\(|\vec{a}+\vec{b}|=\sqrt{37}\)

5. Find the angle between the vectors \(\vec{a}=\hat{i}-\hat{j}+2\hat{k} \,and \,\vec{b}=3\hat{i}+2\hat{j}+4\hat{k}\).
a) \(cos^{-1}⁡\sqrt{\frac{58}{3}}\)
b) \(cos^{-1}⁡\frac{\sqrt{58}}{3}\)
c) \(cos^{-1}\frac{⁡58}{3\sqrt{3}}\)
d) \(cos^{-1}⁡\frac{\sqrt{58}}{3\sqrt{3}}\)
View Answer

Answer: d
Explanation: The angle between the two vectors is given by
\(cos⁡θ=\frac{|\vec{a}|.|\vec{b}|}{\vec{a}.\vec{b}}\)
\(|\vec{a}|=\sqrt{1^2+(-1)^2+2^2}=\sqrt{1+1+4}=\sqrt{6}\)
\(|\vec{b}|=\sqrt{3^2+2^2+(-4)^2}=\sqrt{9+4+16}=\sqrt{29}\)
\(\vec{a}.\vec{b}\)=1(3)-1(2)+2(4)=9
∴\(cos⁡θ=\frac{\sqrt{6}.\sqrt{29}}{9}=\frac{\sqrt{58}}{3\sqrt{3}}\)
∴\(θ=cos^{-1}\frac{⁡\sqrt{58}}{3\sqrt{3}}\)
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6. Find the angle between the vectors \(\vec{a}=-\hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=\hat{i}-\hat{j}\)
a) \(cos^{-1}⁡-\frac{\sqrt{3}}{2}\)
b) \(cos^{-1}⁡-\frac{2}{\sqrt{3}}\)
c) \(cos^{-1}⁡-\sqrt{2}\)
d) \(cos^{-1}⁡-\sqrt{\frac{3}{2}}\)
View Answer

Answer: d
Explanation: The angle between two vectors is given by
\(cos⁡θ=\frac{|\vec{a}|.|\vec{b}|}{\vec{a}.\vec{b}}\)
\(|\vec{a}|=\sqrt{(-1)^2+(1)^2+(-1)^2}=\sqrt{3}\)
\(|\vec{b}|=\sqrt{(1)^2+(-1)^2}=\sqrt{2}\)
\(\vec{a}.\vec{b}\)=(-1)(1)+1(-1)+0=-2
\(cos⁡θ=\frac{\sqrt{3}.\sqrt{2}}{-2}=-\sqrt{\frac{3}{2}}\)
∴\(θ=cos^{-1}⁡-\sqrt{\frac{3}{2}}\)

7. If two non-zero vectors \(\vec{a} \,and \, \vec{b}\) are perpendicular to each other then their scalar product is zero.
a) True
b) False
View Answer

Answer: a
Explanation: The given statement is true. If the angle between two vectors is \(\frac{π}{2}\) i.e. they are perpendicular to each other, their scalar product will be zero.
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8. Find the angle between the two vectors \(\vec{a} \,and \, \vec{b}\) with magnitude 2 and \(\sqrt{3}\) respectively and \(\vec{a.} \, \vec{b}\)=4.
a) \(\frac{π}{3}\)
b) \(\frac{π}{6}\)
c) \(cos^{-1}⁡\frac{\sqrt{2}}{3}\)
d) \(cos^{-1}⁡\frac{2}{\sqrt{3}}\)
View Answer

Answer: b
Explanation: Given that, \(|\vec{a}|=2 \,and \,|\vec{b}|=\sqrt{3}\)
Also, \(\vec{a.} \,\vec{b}=4\)
The angle between two vectors is given by
\(cos⁡θ=\frac{|\vec{a}|.|\vec{b}|}{\vec{a}.\vec{b}}\)
∴\(cos⁡θ=\frac{2.\sqrt{3}}{4}=\frac{\sqrt{3}}{2}\)
∴\(θ=cos^{-1}⁡\frac{\sqrt{3}}{2}=\frac{π}{6}\).

9. Find the scalar product of the vectors \(\vec{a}=6\hat{i}-7\hat{j}+5\hat{k} \,and \,\vec{b}=6\hat{i}-7\hat{k}\)
a) 1
b) 8
c) 6
d) 3
View Answer

Answer: a
Explanation: If \(\vec{a} \,and \,\vec{b}\) are two vectors, where \(a_1, a_2, a_3\) are the components of vector \(\vec{a} \,and \,b_1, b_2, b_3\) are the components of vector \(\vec{b}\), then the scalar product is given by
\(\vec{a}.\vec{b}=a_1 b_1+a_1 b_2+a_3 b_3\)
\((6\hat{i}-7\hat{j}+5\hat{k}).(6\hat{i}-7\hat{k})\)=6(6)-7(0)+5(-7)=36-35=1.
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10. Find the projection of vector \(\vec{b}=2\hat{i}+2\sqrt{2} \,\hat{j}-2\hat{k}\) on the vector \(\vec{a}=\hat{i}-\hat{j}-\sqrt{2} \,\hat{k}\).
a) 2
b) \(\sqrt{2}\)
c) 1
d) \(2\sqrt{2}\)
View Answer

Answer: b
Explanation: The projection of vector \(\vec{b}\) on the vector \(\vec{b}\) is given by \(\frac{1}{|\vec{a}|} (\vec{a}.\vec{b})\)
\(|\vec{a}|=\sqrt{(1)^2+(-1)^2+(-\sqrt{2})^2}=\sqrt{1+1+2}=\sqrt{4}\)=2
Also, \(\vec{a}.\vec{b}=2(1)+2\sqrt{2} \,(-1)-2(-\sqrt{2})=2-2\sqrt{2}+2\sqrt{2}\)=2
Therefore, the projection of vector \(\hat{i}-\hat{j}-\sqrt{2} \,\hat{k}\) on the vector \(\vec{b}=2\hat{i}+2\sqrt{2}\hat{j}-2\hat{k}\) is
\(\frac{1}{|\vec{a}|} (\vec{a}.\vec{b})=\frac{1}{2}\) (2)=1.

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

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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