NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 11 - Three Dimensional Geometry [Latest edition]

If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.

Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).

If a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.

The x-coordinate of a point on the line joining the points Q(2, 2, 1) and R(5, 1, –2) is 4. Find its z-coordinate.

Find the distance of the point whose position vector is `(2hat"i" + hat"j" - hat"k")` from the plane `vec"r" * (hat"i" - 2hat"j" + 4hat"k")` = 9

Find the distance of the point (– 2, 4, – 5) from the line `(x + 3)/3 = (y - 4)/5 = (z + 8)/6`

Find the coordinates of the point where the line through (3, – 4, – 5) and (2, –3, 1) crosses the plane passing through three points (2, 2, 1), (3, 0, 1) and (4, –1, 0)

Find the distance of the point (–1, –5, – 10) from the point of intersection of the line `vec"r" = 2hat"i" - hat"j" + 2hat"k" + lambda(3hat"i" + 4hat"j" + 2hat"k")` and the plane `vec"r" * (hat"i" - hat"j" + hat"k")` = 5

A plane meets the co-ordinates axis in A, B, C such that the centroid of the ∆ABC is the point (α, β, γ). Show that the equation of the plane is `x/alpha + y/beta + z/ϒ` = 3

Find the angle between the lines whose direction cosines are given by the equations: 3l + m + 5n = 0 and 6mn – 2nl + 5lm = 0.

Find the co-ordinates of the foot of perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, –1, 3) and C(2, –3, –1).

Find the image of the point (1, 6, 3) in the line `x/1 = (y - 1)/2 = (z - 2)/3`.

Find the image of the point having position vector `hat"i" + 3hat"j" + 4hat"k"` in the plane `hat"r" * (2hat"i" - hat"j" + hat"k") + 3` = 0.

The coordinates of the foot of the perpendicular drawn from the point (2, 5, 7) on the x-axis are given by ______.

P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.

If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.

The distance of a point P(a, b, c) from x-axis is ______.

The equations of x-axis in space are ______.

A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.

If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.

If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin²α + sin²β + sin²γ is ______.

If a line makes an angle of `pi/4` with each of y and z-axis, then the angle which it makes with x-axis is ______.

The points (1, 2, 3), (–2, 3, 4) and (7, 0, 1) are collinear.

The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`

Find the position vector of a point A in space such that `vec"OA"` is inclined at 60º to OX and at 45° to OY and `|vec"OA"|` = 10 units.

Find the vector equation of the line which is parallel to the vector `3hat"i" - 2hat"j" + 6hat"k"` and which passes through the point (1, –2, 3).

Show that the lines `(x - 1)/2 = (y - 2)/3 = (z - 3)/4` and `(x - 4)/5 = (y - 1)/2` = z intersect. Also, find their point of intersection.

Find the angle between the lines `vec"r" = 3hat"i" - 2hat"j" + 6hat"k" + lambda(2hat"i" + hat"j" + 2hat"k")` and `vec"r" = (2hat"j" - 5hat"k") + mu(6hat"i" + 3hat"j" + 2hat"k")`

Prove that the line through A(0, – 1, – 1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(– 4, 4, 4).

Prove that the lines x = py + q, z = ry + s and x = p′y + q′, z = r′y + s′ are perpendicular if pp′ + rr′ + 1 = 0.

Find the equation of a plane which bisects perpendicularly the line joining the points A(2, 3, 4) and B(4, 5, 8) at right angles.

Find the equation of a plane which is at a distance `3sqrt(3)` units from origin and the normal to which is equally inclined to coordinate axis.

If the line drawn from the point (–2, – 1, – 3) meets a plane at right angle at the point (1, – 3, 3), find the equation of the plane.

Find the equation of the plane through the points (2, 1, 0), (3, –2, –2) and (3, 1, 7).

Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.

Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l² + m² – n² = 0.

If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ² = δl² + δm² + δn² 

O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.

Two systems of rectangular axis have the same origin. If a plane cuts them at distances a, b, c and a′, b′, c′, respectively, from the origin, prove that `1/"a"^2 + 1/"b"^2 + 1/"c"^2 = 1/"a'"^2 + 1/"b'"^2 + 1/"c'"^2`

Find the foot of perpendicular from the point (2, 3, –8) to the line `(4 - x)/2 = y/6 = (1 - z)/3`. Also, find the perpendicular distance from the given point to the line.

Find the distance of a point (2, 4, –1) from the line `(x + 5)/1 = (y + 3)/4 = (z - 6)/(-9)`

Find the length and the foot of perpendicular from the point `(1, 3/2, 2)` to the plane 2x – 2y + 4z + 5 = 0.

Find the equations of the line passing through the point (3, 0, 1) and parallel to the planes x + 2y = 0 and 3y – z = 0.

Find the equation of the plane through the points (2, 1, –1) and (–1, 3, 4), and perpendicular to the plane x – 2y + 4z = 10.

Find the shortest distance between the lines given by `vec"r" = (8 + 3lambdahat"i" - (9 + 16lambda)hat"j" + (10 + 7lambda)hat"k"` and `vec"r" = 15hat"i" + 29hat"j" + 5hat"k" + mu(3hat"i" + 8hat"j" - 5hat"k")`

Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.

The plane ax + by = 0 is rotated about its line of intersection with the plane z = 0 through an angle α. Prove that the equation of the plane in its new position is `"a"x + "b"y +- (sqrt("a"^2 + "b"^2) tan alpha)`z = 0.

Find the equation of the plane through the intersection of the planes `vec"r" * (hat"i" + 3hat"j") - 6` = 0 and `vec"r" * (3hat"i" - hat"j" - 4hat"k")` = 0, whose perpendicular distance from origin is unity.

Show that the points `(hat"i" - hat"j" + 3hat"k")` and `3(hat"i" + hat"j" + hat"k")` are equidistant from the plane `vec"r" * (5hat"i" + 2hat"j" - 7hat"k") + 9` = 0 and lies on opposite side of it.

`vec"AB" = 3hat"i" - hat"j" + hat"k"` and `vec"CD" = -3hat"i" + 2hat"j" + 4hat"k"` are two vectors. The position vectors of the points A and C are `6hat"i" + 7hat"j" + 4hat"k"` and `-9hat"j" + 2hat"k"`, respectively. Find the position vector of a point P on the line AB and a point Q on the line Cd such that `vec"PQ"` is perpendicular to `vec"AB"` and `vec"CD"` both.

Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl + lm = 0 are at right angles.

If l₁, m₁, n₁; l₂, m₂, n₂; l₃, m₃, n₃ are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to l₁ + l₂ + l₃, m₁ + m₂ + m₃, n₁ + n₂ + n₃ makes equal angles with them.

Distance of the point (α, β, γ) from y-axis is ____________.

If the directions cosines of a line are k,k,k, then ______.

The distance of the plane `vec"r" *(2/7hat"i" + 3/4hat"j" - 6/7hat"k")` = 1 from the origin is ______.

The sine of the angle between the straight line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` and the plane 2x – 2y + z = 5 is ______.

The reflection of the point (α, β, γ) in the xy-plane is ______.

The area of the quadrilateral ABCD, where A(0,4,1), B(2, 3, –1), C(4, 5, 0) and D(2, 6, 2), is equal to ______.

The locus represented by xy + yz = 0 is ______.

The plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The value of α is equal to ______.

A plane passes through the points (2, 0, 0) (0, 3, 0) and (0, 0, 4). The equation of plane is ______.

The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.

The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is ______.

The vector equation of the line through the points (3, 4, –7) and (1, –1, 6) is ______.

The Cartesian equation of the plane `vec"r" * (hat"i" + hat"j" - hat"k")` = 2 is ______.

The unit vector normal to the plane x + 2y +3z – 6 = 0 is `1/sqrt(14)hat"i" + 2/sqrt(14)hat"j" + 3/sqrt(14)hat"k"`.

The intercepts made by the plane 2x – 3y + 5z + 4 = 0 on the coordinate axes are `-2, 4/3, (-4)/5`.

The angle between the line `vec"r" = (5hat"i" - hat"j" - 4hat"k") + lambda(2hat"i" - hat"j" + hat"k")` and the plane `vec"r".(3hat"i" - 4hat"j" - hat"k") + 5` = 0 is `sin^-1 (5/(2sqrt(91)))`.

The angle between the planes `vec"r".(2hat"i" - 3hat"j" + hat"k")` = 1 and `vec"r"(hat"i" - hat"j")` = 4 is `cos^-1 ((-5)/sqrt(58))`.

The line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.

The vector equation of the line `(x - 5)/3 = (y + 4)/7 = (z - 6)/2` is `vec"r" = 5hat"i" - 4hat"j" + 6hat"k" + lambda(3hat"i" + 7hat"j" + 2hat"k")`.

The equation of a line, which is parallel to `2hat"i" + hat"j" + 3hat"k"` and which passes through the point (5, –2, 4), is `(x - 5)/2 = (y + 2)/(-1) = (z - 4)/3`.

If the foot of perpendicular drawn from the origin to a plane is (5, – 3, – 2), then the equation of plane is `vec"r".(5hat"i" - 3hat"j" - 2hat"k")` = 38.