Solution of Chapter 11. Three Dimensional Geometry (Mathematics - Important Questions Book)

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 cosines.

Find the vector equation of the line passing through the points (–1, 0, 2) and (3, 4, 6).

Find the vector and Cartesian equations of the line through the point (5, 2, –4) and which is parallel to the vector

Find the angle between the following pairs of lines :

and

Find the shortest distance between the following pairs of lines whose vector equations are :

and

Write the equation of the plane whose intercepts on the coordinate axes are 2, – 3 and 4.

Find the vector and Cartesian equation of the plane which passes through the point (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1.

Find the angle between the planes :

2x + y – 2z = 5 and 3x – 6y – 2z = 7

Find the distance of the point from the plane

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

Find the Cartesian and vector equations of a line which passes through the point (1, 2, 3) and is parallel to the line .

Find the value of so that the following lines are perpendicular to each other.

Reduce the equation to normal form and hence find the length of perpendicular from the origin to the plane.

Find the shortest distance between the following pairs of lines whose Cartesian equations are :

and

Show that the lines and intersect. Find their point of intersection.

Find the vector equation of the plane with intercepts 3, -4 and 2 on x, y and z axes respectively.

Show that the lines and are coplanar.

If the lines and are perpendicular, find the value of k and hence find the equation of the plane containing these lines.

The Cartesian equations of a line are 3x + 1 = 6y – 2 = 1 – z. Find the fixed point through which it passes, its direction ratios and also its vector equation.

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

Find the equation of the line passing through the point (2, 1, 3) and parallel to the lineand

Find the equation of the line passing through the points A(0, 6, – 9) and B( – 3, – 6, 3). If D is the foot of the perpendicular drawn from a point C(7, 4, – 1) on the line AB, then find the coordinates of the point D and the equation of line CD.

Obtain the equation of the plane passing through the point (1, – 3, – 2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes and

Find the distance of the point (1, – 5, 9) from the plane x – y + z = 5 measured along the line x = y = z ?

Find the distance of the point with position vector from the point of intersection of the line with the plane

Find the position vector of the foot of the perpendicular and the perpendicular distance from the point P with position vector to the plane Also, find the image of P in the plane.

Find the equation of the plane through the line of intersection of the plane x + y + z = 1 and 2x + 3y + 4z = 5 and twice of its y–intercept is equals to the three times its z intercept?

Find the vector equation of the plane passing through the intersection of the planes and = – 5 and the point (1, 1, 1).

Find the distance of the point (2, 4, –1) from the line

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

Show that the lines and are coplanar. Hence, find the equation of the plane containing these lines.