Line Direction Cosines & Ratios - Three Dimensional Geometry Questions and Answers

This set of Mathematics Multiple Choice Questions & Answers (MCQs) focuses on “Direction Cosines and Direction Ratios of a Line”.

1. If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is true?
a) \(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ\)
b) \(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ-1\)
c) \(\frac{l}{a}=\frac{m}{c}=\frac{n}{b}=μ\)
d) \(\frac{l}{a}=\frac{n+1}{b}=\frac{n}{c}=μ\)
View Answer

Answer: a
Explanation: For a given line, if a, b, c are the direction ratios and l, m, n are the direction cosines of the line then
a=μl, b=μm, c=μn
Or we can say that,
\(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ\), where μ is a constant.

2. If a line makes an angle of 120°, 45°, 30° with the positive x, y, z-axis respectively then find the direction cosines.
a) l=\(\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)
b) l=-\(\frac{1}{2}, \,m=-\frac{1}{\sqrt{2}}, \,n=-\frac{\sqrt{3}}{2}\)
c) l=-\(\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)
d) l=\(0, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)
View Answer

Answer: c
Explanation: Let l, m, n be the direction cosines of the line.
We know that, if α, β, γ are the angles that the line makes with the x, y, z- axis respectively, then
l=cos⁡α
m=cos⁡β
n=cos⁡γ
∴l=cos⁡120°, m=cos⁡45°, n=cos⁡30°
Hence, \(l=-\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)

3. If a line has direction ratios 2, -3, 7 then find the direction cosines.
a) l=\(\frac{2}{\sqrt{62}},m=-\frac{7}{\sqrt{62}},n=\frac{7}{\sqrt{62}}\)
b) l=\(\frac{2}{\sqrt{6}},m=-\frac{3}{\sqrt{6}},n=\frac{7}{\sqrt{6}}\)
c) l=-\(\frac{2}{\sqrt{62}},m=-\frac{3}{\sqrt{62}},n=-\frac{7}{\sqrt{62}}\)
d) l=\(\frac{2}{\sqrt{62}},m=-\frac{3}{\sqrt{62}},n=\frac{7}{\sqrt{62}}\)
View Answer

Answer: d
Explanation: For a given line, if a, b, c are the direction ratios and l, m, n are the direction cosines of the line then
l=±\(\frac{a}{\sqrt{a^2+b^2+c^2}}\)
m=±\(\frac{b}{\sqrt{a^2+b^2+c^2}}\)
n=±\(\frac{c}{\sqrt{a^2+b^2+c^2}}\)
∴l=\(\frac{2}{\sqrt{2^2+(-3)^2+7^2}}, \,m=-\frac{3}{\sqrt{2^2+(-3)^2+7^2}}, \,n=\frac{7}{\sqrt{2^2+(-3)^2+7^2}}\)
Hence, l=\(\frac{2}{\sqrt{62}}, \,m=-\frac{3}{\sqrt{62}}, \,n=\frac{7}{\sqrt{62}}\).

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4. Find the direction cosines of the line passing through two points (4, -5, -6) and (-1, 2, 8).
a) \(\frac{5}{270},\frac{7}{\sqrt{270}},\frac{14}{\sqrt{270}}\)
b) –\(\frac{7}{\sqrt{270}}, \frac{7}{\sqrt{270}},\frac{7}{\sqrt{270}}\)
c) –\(\frac{5}{\sqrt{270}}, \frac{7}{\sqrt{270}},\frac{14}{\sqrt{270}}\)
d) –\(\frac{5}{\sqrt{20}}, \frac{7}{\sqrt{720}},\frac{14}{\sqrt{270}}\)
View Answer

Answer: c
Explanation: The direction cosines of two lines passing through two points is given by:
\(\frac{x_2-x_1}{PQ}, \frac{y_2-y_1}{PQ}, \frac{z_2-z_1}{PQ} \)
and \(PQ = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)
In the given problem we have, P(4,-5,-6) and Q(-1,2,8)
∴\(PQ = \sqrt{(-1-4)^2+(2+5)^2+(8+6)^2}\)
\(=\sqrt{25+49+196}=\sqrt{270}\)
Hence, the direction ratios are \(l=\frac{(-1-4)}{\sqrt{270}}=-\frac{5}{\sqrt{270}}\)
\(m=\frac{(2+5)}{\sqrt{270}}=\frac{7}{\sqrt{270}}\)
\(n=\frac{(8+6)}{\sqrt{270}}=\frac{14}{\sqrt{270}}\).

5. The direction ratios of the line segment joining \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) is given by _______, ____________ and __________
a) \(x_2+x_1,y_2+y_1,z_2+z_1\)
b) \(x_2-x_1,y_2+y_1,z_2-z_1\)
c) \(x_2-x_1,y_2-y_1,z_2-z_1\)
d) \(x_2+x_1,y_2-y_1,z_2+z_1\)
View Answer

Answer: c
Explanation: Let a.,b,c be the direction ratios of the line segment PQ.
Then, the direction ratios of the line segment joining \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) is given by
\(a=x_2-x_1\)
\(b=y_2-y_1\)
\(c=z_2-z_1\)

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6. Find the direction cosines of the line passing through two points P(-6,7,3) and Q(3,-2,5).
a) –\(\frac{2}{\sqrt{166}},\frac{-9}{\sqrt{166}},\frac{2}{\sqrt{166}}\)
b) –\(\frac{9}{\sqrt{166}},\frac{-7}{\sqrt{166}},\frac{2}{\sqrt{166}}\)
c) –\(\frac{9}{\sqrt{66}},\frac{-9}{\sqrt{66}},\frac{2}{\sqrt{66}}\)
d) –\(\frac{9}{\sqrt{166}},\frac{-9}{\sqrt{166}},\frac{2}{\sqrt{166}}\)
View Answer

Answer: d
Explanation: The direction cosines of two lines passing through two points is given by:
\(\frac{x_2-x_1}{PQ},\frac{y_2-y_1}{PQ},\frac{z_2-z_1}{PQ}\)
and \(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)
In the given problem we have, P(-6,7,3) and Q(3,-2,5)
∴\(PQ=\sqrt{(3+6)^2+(-2-7)^2+(5-3)^2}\)
=\(\sqrt{81+81+4}=\sqrt{166}\)
Hence, the direction ratios are \(l=\frac{-6-3}{\sqrt{166}}=-\frac{9}{\sqrt{166}}\)
m=\(\frac{-2-7}{\sqrt{166}}=\frac{-9}{\sqrt{166}}\)
n=\(\frac{5-3}{\sqrt{166}}=\frac{2}{\sqrt{166}}\)

7. If the direction ratios of a line are 5, 4, -7 respectively, then find the direction cosines.
a) \(\frac{75}{\sqrt{90}},\frac{4}{\sqrt{90}},\frac{5}{\sqrt{90}}\)
b) \(\frac{5}{\sqrt{90}},\frac{4}{\sqrt{90}},-\frac{7}{\sqrt{90}}\)
c) \(\frac{5}{\sqrt{70}},\frac{4}{\sqrt{70}},-\frac{7}{\sqrt{70}}\)
d) \(\frac{3}{\sqrt{90}},\frac{4}{\sqrt{90}},-\frac{5}{\sqrt{90}}\)
View Answer

Answer: b
Explanation: If a,b,c are the direction ratios and l,m,n are the direction cosines respectively for a given line, then the direction cosines in terms of the direction ratios can be expressed as
l=±\(\frac{a}{\sqrt{a^2+b^2+c^2}}\)
m=±\(\frac{b}{\sqrt{a^2+b^2+c^2}}\)
n=±\(\frac{c}{\sqrt{a^2+b^2+c^2}}\)
Given that, a=5, b=4, c=-7
l=\(\frac{5}{\sqrt{(5^2+4^2+(-7)^2)}}=\frac{5}{\sqrt{(25+16+49)}}=\frac{5}{\sqrt{90}}\)
m=\(\frac{4}{\sqrt{(5^2+4^2+(-7)^2)}}=\frac{4}{\sqrt{90}}\)
n=-\(\frac{7}{\sqrt{(5^2+4^2+(-7)^2)}}=-\frac{7}{\sqrt{90}}\)

8. If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is incorrect?
a) \(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k\)
b) l²+m²+n²=1
c) k=±\(\frac{1}{\sqrt{(a^2+b^2+c^2)}}\)
d) l²-m²=n²-1
View Answer

Answer: d
Explanation: Given that, a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line,
\(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=k\) and l²+m²+n²=1
⇒l=ak, m=bk, n=ck
(ak)²+(bk)²+(ck)²=1
k² (a²+b²+c²)=1
k²=\(\frac{1}{a^2+b^2+c^2}\)
∴k=±\(\frac{1}{\sqrt{(a^2+b^2+c^2)}}\)
Hence, l²-m²=n²-1 is incorrect.

9. If the direction cosines of the line are \(\frac{1}{2},-\frac{\sqrt{3}}{2}\),x respectively, then find the value of x.
a) 1
b) 0
c) \(\frac{\sqrt{3}}{2}\)
d) \(\frac{1}{2}\)
View Answer

Answer: b
Explanation: If the direction cosines of a line are l,m,n respectively, then
l²+m²+n²=1
∴\(\frac{1}{2}^2+(\frac{\sqrt{3}}{2})^2+x^2=1\)
x²=\(1-\frac{1}{4}-\frac{3}{4}\)
x²=0
⇒x=0

10. If a line makes an angle of 60°, 150°, 45° with the positive x, y, z-axis respectively, find its direction cosines.
a) –\(\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\)
b) –\(\frac{1}{2},-\frac{\sqrt{3}}{2},-\frac{1}{\sqrt{2}}\)
c) \(\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\)
d) \(\frac{1}{2},\frac{\sqrt{3}}{2},-\frac{1}{\sqrt{2}}\)
View Answer

Answer: c
Explanation: Let l, m, n be the direction cosines of the line.
We know that, if α, β, γ are the angles that the line makes with the x, y, z-axis respectively, then
l=cos⁡α=cos⁡60°=\(\frac{1}{2}\)
m=cos⁡β=cos⁡150°=-\(\frac{\sqrt{3}}{2}\)
n=cos⁡γ=cos⁡45°=\(\frac{1}{\sqrt{2}}\).

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