Mathematics Multiple Choice Questions 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)
Answer: a
Clarification: 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})
Answer: c
Clarification: 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=cosα
m=cosβ
n=cosγ
∴l=cos120°, m=cos45°, n=cos30°
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}})
Answer: d
Clarification: 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}}).
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}})
Answer: c
Clarification: 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)
Answer: c
Clarification: 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)
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}})
Answer: d
Clarification: 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}})
Answer: b
Clarification: 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