# Mathematics Questions and Answers – Direction Cosines and Direction Ratios of a Line

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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.
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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=-$$\frac{7}{\sqrt{(5^2+4^2+(-7)^2)}}=-\frac{7}{\sqrt{90}}$$
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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
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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°=$$\frac{1}{\sqrt{2}}$$.

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