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Chapter 4 : Three Dimensional Geometry
Q. If a line makes angles 90, 135, 45 with the x, y and Zaxes respectively, find its direction cosines.
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Q. Find the direction cosines of a line which makes equal angles with the coordinate axes.
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Q. If a line has the direction ratios 18, 12, 4, then what are its direction cosines?
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Q. Find the values of p so the line 1x3=7y142p=z32 and 77x3p=y51=6z5 are at right angles.
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Q. Find the vector and the Cartesian equations of the lines that pass through the origin and (5, 2, 3)
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Q. Find the shortest distance between the lines whose vector equations are r=(^i+2^j+3^k)+λ(^i3^j+2^k) and r=(4^i+5^j+6^k)+μ(2^i+3^j+^k).
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Q. Find the Cartesian equation of the line which passes through the point (2, 4, 5) and parallel to the line given by x+33=y45=z+86
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Q. Find the equation of the line in vector and in cartesian form that passes through the point with position vector 2^i^j+4^k and is in the direction ^i+2^j^k.
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Q. The cartesian equation of a line is x53=y+47=z62 write its vector form.
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Q. Find the vector and the Cartesian equations of the line that passes through the points (3, 2, 5), (3, 2, 6)
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Q. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector (3^i+2^j2^k.
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Q. x+y+z=1
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Q. Find the shortest distance between the lines
r=(^i+2^j+^k)+λ(^i^j+^k) and
r=^2i^j^k+μ(^2i+^j+2^k)
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Q. Find the angle between the following pair of lines:
(i) r=2^i5^j+^k+λ(3^i2^j+6^k) and r=7^i6^k+μ(^i+2^j+2^k)
(ii) r=3^i+^j2^k+λ(^i^j2^k) and r=2^i^j56^k+μ(^3i5^j4^k)
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Q. Find the shortest distance between the lines whose vector equations are :
r=(1p)^i+(p2)^j+(32p)^k
r=(q+1)^i+(2q1)^j(2q1)^k
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Q. Find the shortest distance between lines x+17=y+16=z+11 and x31=y52=z71
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Q. The angle between the following pair of lines:
x2=y2=z1 and x54=y21=z38 is θ=cos1a3 then a=
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Q.
In the following case, determine the direction cosines of the normal to the plane and the distance from the origin.
(i) z=2
(ii) x+y+z=1
(iii) 2x+3y5=0
(iv) 5y+8=0
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Q. Find the angle between the following pairs of lines:
r=3^i+^j2^k+λ(^i+^j2^k) and r=2^i^j56^k+μ(3^i5^j4^k).
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Q. Find the angle between the following pair of lines:
(i) x22=y15=z+33 and x+21=y48=z54
(ii) x2=y2=z1 and x54=y21=z38
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