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

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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
\(cos⁡θ=\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
\(θ=cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)
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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
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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
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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=\(\left |\frac{\vec{b_1}.\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}\right |\).
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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=\(\left |l_1 \,l_2+m_1 \,m_2+n_1 \,n_2\right |\)
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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.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He lives in Bangalore, and focuses on development of Linux Kernel, SAN Technologies, Advanced C, Data Structures & Alogrithms. Stay connected with him at LinkedIn.

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