Mathematics Questions and Answers – Three Dimensional Geometry – Angle between Two Lines

«
»

This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Three Dimensional Geometry – Angle between Two Lines”.

1. If L1 and L2 have the direction ratios \(a_1,b_1,c_1 \,and \,a_2,b_2,c_2\) respectively then what is the angle between the lines?
a) \(θ=tan^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
b) \(θ=2tan^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
c) \(θ=cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
d) \(θ=2 \,cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
View Answer

Answer: c
Explanation: If L1 and L2 have the direction ratios \(a_1,b_1,c_1 \,and \,a_2,b_2,c_2\) respectively then the angle between the lines is given by
\(cos⁡θ=\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
\(θ=cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
advertisement

2. Find the angle between the lines.
\(\frac{x+2}{1}=\frac{y+5}{6}=\frac{z-3}{2}\)
\(\frac{x-4}{5}=\frac{y-3}{-2}=\frac{z+3}{1}\)
a) \(cos^{-1}\frac{⁡5}{\sqrt{1230}}\)
b) \(cos^{-1}⁡\frac{⁡3}{\sqrt{3120}}\)
c) \(cos^{-1}⁡\frac{⁡7}{\sqrt{2310}}\)
d) \(cos^{-1}\frac{⁡⁡48}{\sqrt{1230}}\)
View Answer

Answer: a
Explanation: We know that, the angle between two lines is given by the formula
cos⁡θ=\(\left |\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
cos⁡θ=\(\left |\frac{1(5)+6(-2)+2(1)}{\sqrt{1^2+6^2+2^2).√(5^2+(-2)^2+1^2}}\right |\)
=\(\left |\frac{-5}{\sqrt{41}.\sqrt{30}} \right |=\frac{5}{\sqrt{1230}}\)
∴\(θ=cos^{-1}\frac{5}{\sqrt{1230}}\)

3. Find the value of p such that the lines
\(\frac{x-1}{3}=\frac{y+4}{p}=\frac{z-9}{1}\)
\(\frac{x+2}{1}=\frac{y-3}{1}=\frac{z-7}{-2}\)
are at right angles to each other.
a) p=2
b) p=1
c) p=-1
d) p=-2
View Answer

Answer: c
Explanation: The angle between two lines is given by the equation
\(cos⁡θ=\left |\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
cos⁡90°=\(\left |\frac{3(1)+p(1)+1(-2)}{\sqrt{3^2+p^2+1^2}.\sqrt{1^2+1^2+(-2)^2}}\right |\)
0=\(|\frac{p+1}{\sqrt{10+p^2}.√6}|\)
0=p+1
p=-1
advertisement
advertisement

4. Find the angle between the two lines if the equations of the lines are
\(\vec{r}=\hat{i}+\hat{j}+\hat{k}+λ(3\hat{i}-\hat{j}+\hat{k}) \,and \,\vec{r}=4\hat{i}+\hat{j}-2\hat{k}+μ(2\hat{i}+3\hat{j}+\hat{k})\)
a) \(cos^{-1}\frac{⁡4}{\sqrt{14}}\)
b) \(cos^{-1}⁡\frac{7}{\sqrt{154}}\)
c) \(cos^{-1}⁡\frac{4}{154}\)
d) \(cos^{-1}⁡\frac{4}{\sqrt{154}}\)
View Answer

Answer: d
Explanation: Given that, \(\vec{r}=\hat{i}+\hat{j}+\hat{k}+λ(3\hat{i}-\hat{j}+\hat{k})\) and \(\vec{r}=4\hat{i}+\hat{j}-2\hat{k}+μ(2\hat{i}+3\hat{j}+\hat{k})\)
We know that, if the equations of two lines are of the form \(\vec{r}=\vec{a_1}+λ\vec{b_1} and \,\vec{r}=\vec{a_2}+μ\vec{b_2}\) then the angle between the two lines is given by
\(cos⁡θ=\left|\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right|\)
=\(\left |\frac{(3(2)+(-1)3+1(1)}{\sqrt{3^2+(-1)^2+(1)^2.} \sqrt{2^2+3^2+1^2}}\right |=\frac{4}{\sqrt{11}.\sqrt{14}}=\frac{4}{\sqrt{154}}\)
θ=\(cos^{-1}⁡\frac{4}{\sqrt{154}}\).

5. If two lines L1 and L2 with direction ratios \(a_1,b_1,c_1 \,and \,a_2,b_2,c_2\) respectively are perpendicular to each other then
\(a_1 a_2+b_1 b_2+c_1 c_2=0\)
a) True
b) False
View Answer

Answer: a
Explanation: The given statement is true.
We know that the angle between two lines is given by the formula
cos⁡θ=\(\left |\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
So, if the lines L1 and L2 are perpendicular to each to each other then,
θ=90°
⟹\(a_1 \,a_2+b_1 \,b_2+c_1 \,c_2\)=0
advertisement

6. Find the value of p such that the lines \(\frac{x+11}{4}=\frac{y+3}{-2}=\frac{z-3}{4} \,and \,\frac{x-3}{p}=\frac{y+12}{2}=\frac{z-3}{-12}\) are at right angles to each other.
a) p=11
b) p=12
c) p=13
d) p=4
View Answer

Answer: c
Explanation: We know that, if two lines are perpendicular to each other then,
\(a_1 a_2+b_1 b_2+c_1 c_2=0\)
i.e.4(p)+(-2)2+4(-12)=0
4p-4-48=0
4p=52
p=\(\frac{52}{4}\)=13.

7. If the equations of two lines L1 and L2 are \(\vec{r}=\vec{a_1}+λ\vec{b_1}\) and \(\vec{r}=\vec{a_2}+μ\vec{b_2}\), then which of the following is the correct formula for the angle between the two lines?
a) cos⁡θ=\(\left |\frac{\vec{a_1}.\vec{a_2}}{|\vec{b_1}||\vec{a_2}|}\right |\)
b) cos⁡θ=\(\left |\frac{\vec{a_1}.\vec{a_2}}{|\vec{a_1}||\vec{a_2}|}\right |\)
c) cos⁡θ=\(\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |\)
d) cos⁡θ=\(\left |\frac{\vec{a_1}.\vec{b_2}}{|\vec{a_1}||\vec{b_2}|}\right |\)
View Answer

Answer: c
Explanation: Given that the equations of the lines are
\(\vec{r}=\vec{a_1}+λ\vec{b_1} \,and \,\vec{r}=\vec{a_2}+μ\vec{b_2}\)
∴ the angle between the two lines is given by
cos⁡θ=\(\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |\).
advertisement

8. Find the angle between the lines \(\vec{r}=2\hat{i}+6\hat{j}-\hat{k}+λ(\hat{i}-2\hat{j}+3\hat{k})\) and \(\vec{r}=4\hat{i}-7\hat{j}+3\hat{k}+μ(5\hat{i}-3\hat{j}+3\hat{k})\).
a) θ=\(cos^{-1}\frac{⁡20}{\sqrt{602}}\)
b) θ=\(cos^{-1}\frac{⁡20}{\sqrt{682}}\)
c) θ=\(cos^{-1}\frac{⁡8}{\sqrt{602}}\)
d) θ=\(cos^{-1}⁡\frac{14}{\sqrt{598}}\)
View Answer

Answer: a
Explanation: If two lines have the equations \(\vec{r}=\vec{a_1}+λ\vec{b_1} \,and \,\vec{r}=\vec{a_2}+μ\vec{b_2}\)
Then, the angle between the two lines will be given by
cos⁡θ=\(\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |\)
=\(\left |\frac{(\hat{i}-2\hat{j}+3\hat{k}).(5\hat{i}-3\hat{j}+3\hat{k})}{\sqrt{1^2+(-2)^2+(3)^2).√(5^2+(-3)^2+3^2}}\right |\)
=\(\frac{5+6+9}{√14.√43}=\frac{20}{√602}\)
θ=\(cos^{-1}⁡\frac{20}{\sqrt{602}}\)

9. If two lines L1 and L2 are having direction cosines \(l_1,m_1,n_1 \,and \,l_2,m_2,n_2\) respectively, then what is the angle between the two lines?
a) cot⁡θ=\(\left |l_1 \,l_2+m_1 \,m_2+n_1 \,n_2\right |\)
b) sin⁡θ=\(\left |l_1 \,l_2+m_1 \,n_2+n_1 \,m_2\right |\)
c) tan⁡θ=\(\left |l_1 \,l_2+m_1 \,m_2+n_1 \,n_2\right |\)
d) cos⁡θ=\(\left |l_1 \,l_2+m_1 \,m_2+n_1 \,n_2\right |\)
View Answer

Answer: d
Explanation: If two lines L1 and L2 are having direction cosines \(l_1,m_1,n_1 \,and \,l_2,m_2,n_2\) respectively, then the angle between the lines is given by
cos⁡θ=\(\left |l_1 \,l_2+m_1 \,m_2+n_1 \,n_2\right |\)
advertisement

10. Find the angle between the pair of lines \(\frac{x-3}{5}=\frac{y+7}{3}=\frac{z-2}{2} \,and \,\frac{x+1}{3}=\frac{y-5}{4}=\frac{z+2}{8}\).
a) \(cos^{-1}⁡\frac{43}{\sqrt{3482}}\)
b) \(cos^{-1}⁡⁡\frac{43}{\sqrt{3382}}\)
c) \(cos^{-1}⁡⁡\frac{85}{\sqrt{3382}}\)
d) \(cos^{-1}⁡⁡\frac{34}{\sqrt{3382}}\)
View Answer

Answer: b
Explanation: The direction ratios are 5, 3, 2 for L1 and 3, 4, 8 for L2
∴ the angle between the two lines is given by
cos⁡θ=\(\frac{(a_1 a_2+b_1 b_2+c_1 c_2)}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\)
=\(\frac{15+12+16}{\sqrt{5^2+3^2+2^2}.\sqrt{3^2+4^2+8^2}}\)
=\(\frac{43}{\sqrt{38}.\sqrt{89}}=\frac{43}{\sqrt{3382}}\)
θ=\(cos^{-1}⁡\frac{43}{\sqrt{3382}}\).

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