If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines.

Find the direction cosines of a line which makes equal angles with the coordinate axes.

If a line has the direction ratios -18, 12, -4, then what are its direction cosines?

Show that the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) 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)

Show that the three lines with direction cosines are mutually perpendicular.

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

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).

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to thevector.

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 .

Find the Cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by

The Cartesian equation of a line is . Write its vector form.

Find the vector and the Cartesian equations of the lines that pass through the origin and (5, -2, 3).

Find the vector and the Cartesian equations of the line that passes through the points (3, -2, -5), (3, -2, 6).

Find the angle between the following pairs of lines: (i) (ii)and

Find the angle between the following pairs of lines: (i) (ii)

Find the values of p sothelineand are at right angles.

Show that the lines and are perpendicular to each other.

Find the shortest distance between the lines

Find the shortest distance between the lines and

Find the shortest distance between the lines whose vector equations are

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

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector .

Find the Cartesian equation of the following planes: (a)(b) (c)

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin. (a)(b) (c)(d)

Find the vector and Cartesian equation of the planes (I) thatpassesthroughthepoint(1,0,-2)andthenormaltotheplaneis. (II) that passes through the point (1, 4, 6) and the normal vector to the planeis .

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)

Find the intercepts cut off by the plane

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Find the equation of the plane through the intersection of the planes andand the point (2, 2,1)

Find the vector equation of the plane passing through the intersection of the planes and through the point (2, 1, 3)

Find the equation of the plane through the line of intersection of the planes and which is perpendicular to theplane

Find the angle between the planes whose vector equations are and.

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) (I) (II)

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)

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).

If l₁, m₁, n₁ and l₂, m₂, n₂ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m₁n₂ - m₂n₁, n₁l₂ - n₂l₁, l₁m₂ - l₂m₁.

Findtheanglebetweenthelineswhosedirectionratiosarea,b,candb-c, c - a, a -b.

Find the equation of a line parallel to x-axis and passing through the origin.

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.

Ifthelinesandare perpendicular, find thevalue ofk.

Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane