If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.
Answer :
Let direction cosines of the line be l, m, and n.
Therefore, the direction cosines of the line are
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Find the direction cosines of a line which makes equal angles with the coordinate axes.
Answer :
Let the direction cosines of the line make an angle α with each of the coordinate axes.
∴ l= cos α, m = cos α, n = cos α
Thus, the direction cosines of the line, which is equally inclined to the coordinate axes, are
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If a line has the direction ratios -18, 12, -4, then what are its direction cosines?
Answer :
Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) are collinear.
Answer :
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, - 4), (- 1, 1, 2) and (- 5, - 5, - 2)
Answer :
Show that the three lines with direction cosines
are mutually perpendicular.
Answer :
Show that the line through the points (1, -1, 2) (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Answer :
Show that the line through the points (4, 7, 8) (2, 3, 4) is parallel to the line through the points (-1, -2, 1), (1, 2, 5).
Answer :
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector.
Answer :
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector and is in the direction .
Answer :
Find the Cartesian equation of the line which passes through the point
( - 2, 4, - 5) and parallel to the line given by
Answer :
The Cartesian equation of a line is . Write its vector form.
Answer :
Find the vector and the Cartesian equations of the lines that pass through the origin and (5, -2, 3).
Answer :
Find the vector and the Cartesian equations of the line that passes through the points (3, -2, -5), (3, -2, 6).
Answer :
Find the angle between the following pairs of lines:
(i)
(ii) and
Answer :
Find the angle between the following pairs of lines:
(i)
(ii)
Answer :
Find the values of p so the line and
are at right angles.
Answer :
Show that the lines and are perpendicular to each other.
Answer :
Find the shortest distance between the lines
Answer :
Find the shortest distance between the lines and
Answer :
Find the shortest distance between the lines whose vector equations are
Answer :
Find the shortest distance between the lines whose vector equations are
Answer :
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a)z = 2 (b)
(c) (d)5y + 8 = 0
Answer :
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector.
Answer :
Find the Cartesian equation of the following planes:
(a) (b)
(c)
Answer :
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) (b)
(c) (d)
Answer :
Find the vector and Cartesian equation of the planes
(a) that passes through the point (1, 0, - 2) and the normal to the plane is .
(b) that passes through the point (1, 4, 6) and the normal vector to the plane is .
Answer :
Find the equations of the planes that passes through three points.
(a) (1, 1, -1), (6, 4, -5), (-4, -2, 3)
(b) (1, 1, 0), (1, 2, 1), (-2, 2, -1)
Answer :
Find the intercepts cut off by the plane
Answer :
Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.
Answer :
Find the equation of the plane through the intersection of the planes and and the point (2, 2, 1)
Answer :
Find the vector equation of the plane passing through the intersection of the planes and through the point (2, 1, 3)
Answer :
Find the equation of the plane through the line of intersection of the planes and which is perpendicular to the plane
Answer :
Find the angle between the planes whose vector equations are
and .
Answer :
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a)
(b)
(c)
(d)
(e)
Answer :
In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(a) (0, 0, 0)
(b) (3, - 2, 1)
(c) (2, 3, - 5)
(d) ( - 6, 0, 0)
Answer :
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, -1), (4, 3, -1).
Answer :
If l_{1}, m_{1}, n_{1} and l_{2}, m_{2}, n_{2} are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_{1}n_{2} - m_{2}n_{1}, n_{1}l_{2} - n_{2}l_{1}, l_{1}m_{2} - l_{2}m_{1}.
Answer :
Find the angle between the lines whose direction ratios are a, b, c and b - c,
c - a, a - b.
Answer :
Find the equation of a line parallel to x-axis and passing through the origin.
Answer :
If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (-4, 3, -6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.
Answer :
If the lines and are perpendicular, find the value of k.
Answer :
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane
Answer :
Find the equation of the plane passing through (a, b, c) and parallel to the plane
Answer :
Find the shortest distance between lines
and.
Answer :
Find the coordinates of the point where the line through (5, 1, 6) and
(3, 4, 1) crosses the YZ-plane
Answer :
Find the coordinates of the point where the line through (5, 1, 6) and
(3, 4, 1) crosses the ZX - plane.
Answer :
Find the coordinates of the point where the line through (3, -4, -5) and (2, - 3, 1) crosses the plane 2x + y + z = 7).
Answer :
Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
Answer :
If the points (1, 1, p) and ( - 3, 0, 1) be equidistant from the plane , then find the value of p.
Answer :
Find the equation of the plane passing through the line of intersection of the planes and and parallel to x-axis.
Answer :
If O be the origin and the coordinates of P be (1, 2, -3), then find the equation of the plane passing through P and perpendicular to OP.
Answer :
Find the equation of the plane which contains the line of intersection of the planes , and which is perpendicular to the plane .
Answer :
Find the distance of the point ( - 1, - 5, - 10) from the point of intersection of the line and the plane.
Answer :
Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and .
Answer :
Find the vector equation of the line passing through the point (1, 2, - 4) and perpendicular to the two lines:
Answer :
Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then
Answer :
Distance between the two planes: and is
(A)2 units (B)4 units (C)8 units
(D)
Answer :
The planes: 2x - y + 4z = 5 and 5x - 2.5y + 10z = 6 are
(A) Perpendicular (B) Parallel (C) intersect y-axis
(C) passes through
Answer :
Maths Part-2 - Maths : CBSE NCERT Exercise Solutions for Class 12th for Three Dimensional Geometry ( Exercise 11.1, 11.2, 11.3, miscellaneous ) will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.
Exercise 11.1 |
Question 1 |
Question 2 |
Question 3 |
Question 4 |
Question 5 |
Exercise 11.3 |
Question 1 |
Question 2 |
Question 3 |
Question 4 |
Question 5 |
Question 6 |
Question 7 |
Question 8 |
Question 9 |
Question 10 |
Question 11 |
Question 12 |
Question 13 |
Question 14 |