RD Sharma solutions for Mathematics [English] Class 12 chapter 27 - Direction Cosines and Direction Ratios [Latest edition]

If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines

If a line has direction ratios 2, −1, −2, determine its direction cosines.

Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .

Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).

Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.

Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.

Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).

Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

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) and (4, 3, −1).

Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.

If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.

Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.

Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l² + m² − n² = 0

Find the angle between the lines whose direction cosines are given by the equations

2l − m + 2n = 0 and mn + nl + lm = 0

Find the angle between the lines whose direction cosines are given by the equations

 l + 2m + 3n = 0 and 3lm − 4ln + mn = 0

Find the angle between the lines whose direction cosines are given by the equations

2l + 2m − n = 0, mn + ln + lm = 0

Define direction cosines of a directed line.

What are the direction cosines of X-axis?

What are the direction cosines of Y-axis?

What are the direction cosines of Z-axis?

Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.

Write the distance of the point (3, −5, 12) from X-axis?

Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).

A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.

If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.

Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.

Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

Write the distance of the point P (x, y, z) from XOY plane.

Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.

Write the coordinates of the projection of the point P (2, −3, 5) on Y-axis.

Find the distance of the point (2, 3, 4) from the x-axis.

If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?

Write direction cosines of a line parallel to z-axis.

If a unit vector  `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with }  \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.

Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(a, b, c) from x-axis.

If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.

For every point P (x, y, z) on the xy-plane,

For every point P (x, y, z) on the x-axis (except the origin),

A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is

A parallelopiped is formed by planes drawn through the points (2, 3, 5) and (5, 9, 7), parallel to the coordinate planes. The length of a diagonal of the parallelopiped is

The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)

If the x-coordinate of a point P on the join of Q (2, 2, 1) and R (5, 1, −2) is 4, then its z-coordinate is

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

Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio

If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are

The angle between the two diagonals of a cube is

If a line makes angles α, β, γ, δ with four diagonals of a cube, then cos² α + cos² β + cos²γ + cos² δ is equal to