NCERT Exemplar Solutions
Class 11 - Mathematics

Chapter : INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

In which octant do the points lie?

Find the coordinates of the feet of perpendiculars from a point P on the xy, yz and zx-planes respectively for the given cases.

Find how far apart the points (2, 0, 0) and (−3, 0, 0) are.

Find the distance from the origin to the point (6, 6, 7).

Show that if \(x^2 + y^2 = 1\), then the point \((x, y, \sqrt{1 - x^2 - y^2})\) is at a distance 1 unit from the origin.

Show that the points A (1, −1, 3), B (2, −4, 5) and (5, −13, 11) are collinear.

Three consecutive vertices of a parallelogram ABCD are A (6, −2, 4), B (2, −4, −8), and C (−2, 2, 4). Find the coordinates of the fourth vertex.

Show that the triangle ABC with vertices A (0, 4, 1), B (2, 3, −1) and C (4, 5, 0) is right angled.

Find the third vertex of a triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, −2, −5).

Find the centroid of a triangle the mid-point of whose sides are given as D (1, 2, −3), E (3, 0, 1), and F (−1, 1, −4).

The mid-points of the sides of a triangle are (5, 7, 11), (0, 8, 5), and (2, 3, −1). Find its vertices.

Three vertices of a parallelogram ABCD are A (1, 2, 3), B (−1, −2, −1), and C (2, 3, 2). Find the fourth vertex D.

Find the coordinates of the points which trisect the line segment joining A (2, 1, −3) and B (5, −8, 3).

If the origin is the centroid of a triangle ABC having vertices A (a, 1, 3), B (−2, b, −5), and C (4, 7, c), find the values of a, b, c.

Let A (2, 2, −3), B (5, 6, 9) and C (2, 7, 9) be the vertices of a triangle. The internal bisector of angle A meets BC at point D. Find the coordinates of D.

Show that the three points A (2, 3, 4), B (−1, 2, −3) and C (−4, 1, −10) are collinear and find the ratio in which C divides AB.

The distance of point P (3, 4, 5) from the yz-plane is?

Show that the three points A (2, 3, 4), B (−1, 2, −3) and C (−4, 1, −10) are collinear and find the ratio in which C divides AB.

The mid-point of the sides of a triangle are (1, 5, −1), (0, 4, −2) and (2, 3, 4). Find its vertices. Also find the centroid of the triangle.

Prove that the points (0, −1, −7), (2, 1, −9) and (6, 5, −13) are collinear. Find the ratio in which the first point divides the join of the other two.

What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin coincide with the positive direction of the axes through the origin?

The three axes OX, OY, OZ determine ____.

The three planes determine a rectangular parallelepiped which has ______ of rectangular faces.

The coordinates of a point are the perpendicular distance from the ______ on the respective axes.

The three coordinate planes divide the space into ______ parts.

If a point P lies in yz-plane, then the coordinates of a point on yz-plane is of the form ______.

The equation of yz-plane is ______.

If the point P lies on z-axis, then coordinates of P are of the form ______.

The equation of z-axis, are ______.

A line is parallel to xy-plane if all the points on the line have equal ______.

A line is parallel to x-axis if all the points on the line have equal ______.

x = a represent a plane parallel to ______.

The plane parallel to yz-plane is perpendicular to ______.

The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10, 13 and 8 units are ______.

If the distance between the points (a, 2, 1) and (1, –1, 1) is 5, then a ______.

If the mid-points of the sides of a triangle AB; BC; CA are D (1, 2, –3), E (3, 0, 1) and F (–1, 1, –4), then the centroid of the triangle ABC is ______.

Match the items in Column A with Column B using the table below.