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Name the octants in which the following points lie:

(i) (5, 2, 3)

(ii) (-5, 4, 3)

(iii) (4, -3, 5)

(iv) (7, 4, -3)

(v) (-5, -4, 7)

(vi) (-5, -3, -2)

(vii) (2, -5, -7)

(viii) (-7, 2, -5)

Find the image of:

(i) (-2, 3, 4) in the yz-plane

(ii) (-5, 4, -3) in the xz-plane

(iii) (5, 2, -7) in the xy-plane

(iv) (-5, 0, 3) in the xz-plane

(v) (-4, 0, 0) in the xy-plane

A cube of side 5 has one vertex at the point (1, 0, 1), and the three edges from this vertex are, respectively, parallel to the negative x and y-axes and positive z-axis. Find the coordinates of the other vertices of the cube.

Planes are drawn parallel to the coordinates planes through the points (3, 0, -1) and (-2, 5, 4). Find the lengths of the edges of the parallelepiped so formed.

Planes are drawn through the points (5, 0, 2) and (3, -2, 5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelepiped so formed.

Find the distances of the point P (-4, 3, 5) from the coordinate axes.

The coordinates of a point are (3, -2, 5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.

Find the distance between the following pairs of points :

P(1, -1, 0) and Q (2, 1, 2)

A(3, 2, -1) and B (-1, -1, -1)

Find the distance between the points P and Q having coordinates (-2, 3, 1) and (2, 1, 2).

Using distance formula prove that the following points are collinear :

A(4, -3, -1), B(5, -7, 6) and C(3, 1, -8)

P(0, 7, -7), Q(1, 4, -5) and R(-1, 10, -9)

A(3, -5, 1), B(-1, 0, 8) and C(7, -10, -6)

Determine the points in (i) xy-plane (ii) yz-plane and (iii) zx-plane which are equidistant from the points A(1, -1, 0), B(2, 1, 2) and C(3, 2, -1).

Determine the point on z-axis which is equidistant from the points (1, 5, 7) and (5, 1, -4)

Find the point on y-axis which is equidistant from the points (3, 1, 2) and (5, 5, 2).

Find the points on z-axis which are at a distance from the point (1, 2, 3).

Prove that the triangle formed by joining the three points whose coordinates are (1, 2, 3), (2, 3, 1) and (3, 1, 2) is an equilateral triangle.

Show that the points (0, 7, 10), (-1, 6, 6) and (-4, 9, 6) are the vertices of an isosceles right-angled triangle.

Show that the points A(3, 3, 3), B(0, 6, 3), C(1, 7, 7) and D(4, 4, 7) are the vertices of squares.

Prove that the point A(1, 3, 0), B(-5, 5, 2), C(-9, -1, 2) and D(-3, -3, 0) taken in order are the vertices of a parallelogram. Also, show that ABCD is not a rectangle.

Show that the points A(1, 3, 4), B(-1, 6, 10), C(-7, 4, 7) and D(-5, 1, 1) are the vertices of a rhombus.

Prove that the tetrahedron with vertices at the points O(0, 0, 0), A(0, 1, 1), B(1, 0, 1) and C(1, 1, 0) is a regular one.

Show that the points (3, 2, 2), (-1, 4, 2), (0, 5, 6), (2, 1, 2) lie on a sphere whose centre is (1, 3, 4). Find also its radius.

Find the coordinates of the point which is equidistant from the four points O(0, 0, 0), A(2, 0, 0), B(0, 3, 0) and C(0, 0, 8).

If A(-2, 2, 3) and B(13, -3, 13) are two pints. Find the locus of a point P which moves in such a way that 3PA = 2PB.

Find the locus of P if PA^{2} + PB^{2} = 2k^{2}, where A and B are the points (3, 4, 5) and (-1, 3, -7).

Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.

Are the points A(3, 6, 9), B(10, 20, 30) and C(25, -41, 5), the vertices of a right-angled triangle?

Verify the following:

(0, 7, -10), (1, 6, -6) and (4, 9, -6) are vertices of an isosceles triangle.

(0, 7, 10), (-1, 6, 6) and (2, -3, 4) are vertices of a right-angled triangle

(-1, 2, 1), (1,-2, 5), (4, -7, 8) and (2, -3, 4) are vertices of a parallelogram.

(5, -1, 1), (7, -4, 7), (1, -6, 10) and (-1, -3, 4) are the vertices of a rhombus.

Find the locus of the points which are equidistant from the points (1, 2, 3) and (3, 2, -1).

Find the locus of the point, the sum of whose distances from the points A(4, 0, 0) and B(-4, 0, 0) is equal to 10.

Show that the point A(1,2, 3), B(-1, -2, -1), C(2, 3, 2) and D(4, 7, 6) are the vertices of a parallelogram ABCD, but not a rectangle.

Find the equation of the set of the points P such that its distances from the points A(3, 4, -5) and B(-2, 1, 4) are equal.

The vertices of the triangle are A(5, 4, 6), B(1, -1, 3) and C(4, 3, 2). The internal bisector of angle A meets BC at D. Find the coordinates of D and the length AD.

A point C with z-coordinate 8 lies on the line segment joining the points A(2, -3, 4) and B(8, 0, 10). Find the coordinates.

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.

Find the ratio in which the line joining (2, 4, 5) and (3, 5, 4) is divided by the yz-plane.

Find the ratio in which the line segment joining the points (2, -1, 3) and (-1, 2, 1) is divided by the plane x + y + z = 5.

If the points A(3, 2, -4), B(9, 8, -10) and C(5, 4, -6) are collinear, find the ratio in which C divided AB.

The mid-points of the sides of a triangle ABC are given by (-2, 3, 5), (4, -1, 7) and (6, 5, 3). Find the coordinates of A, B and C.

A(1, 2, 3), B(0, 4, 1), C(-1, -1, -3) are the vertices of a triangle ABC. Find the point in which the bisector of the angle ∠BAC meets BC.

Find the ratio in which the sphere _{x}^{2} + y^{2} + z^{2} = 504 divides the line joining the point (12, -4, 8) and (27, -9, 18).

Show that the plane ax + by + cz + d = 0 divides the line joining the points (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) in the ratio .

Find the centroid of a triangle, mid-points of whose are (1, 2, -3), (3, 0, 1) and (-1, 1, -4).

The centroid of a triangle ABC is at the point (1, 1, 1). If the coordinates of A and B are (3, -5, 7) and (-1, 7, -6) respectively, find the coordinates of the point C.

Find the coordinates of the points which trisect the line segment joining the points P(4, 2, -6) and Q(10, -16, 6).

Using section formula, show that the points A(2, -3, 4), B(-1, 2, 1) and C(0, 1/3, 2) are collinear.

Given that P(3, 2, -4), Q(5, 4, -6) and R(9, 8, -10) are collinear. Find the ratio in which Q divides PR.

Find the ratio in which the line segment joining the points (4, 8, 10) and (6, 10, -8) is divided by the yz-plane.

Write the distance of the point P(2, 3, 5) from the xy-plane.

Write the distance of the point P(3, 4, 5) from the z-axis.

If the distance between the points P(a, 2, 1) and Q(1, -1, 1) is 5 units, find the value of a.

The coordinates of the mid-points of sides AB, BC and CA of ΔABC are D(1, 2, -3), E(3, 0, 1) and F(-1, 1, -4) respectively. Write the coordinates of its centroid.

Write the coordinates of the foot of the perpendicular from the point P(1, 2, 3) on the y-axis.

Write the length of the perpendicular drawn from the point P(3, 5, 12) on the x-axis.

Write the coordinates of the third vertex of a triangle having centroid at the origin and two vertices at (3, -5, 7) and (3, 0, 1).

What is the locus of a point (x, y, z) for which y = 0, z = 0?

Find the ratio in which the line segment joining the points (2, 4, 5) and (3, -5, 4) is divided by the yz-plane.

Find the point on y-axis which is at a distance of units from the point (1, 2, 3).

Find the point on x-axis which is equidistant from the points A(3, 2, 2) and B(5, 5, 4).

Find the coordinates of a point equidistant from the origin and points A(a, 0, 0), B(0, b, 0) and C(0, 0, c).

Write the coordinates of the point P which is five-sixth of the way from A(-2, 0, 6) to B(10, -6, -12).

If a parallelepiped is formed by the planes drawn through the points (2, 3, 5) and (5, 9, 7) parallel to the coordinates planes, then write the lengths of edges of the parallelepiped and length of the diagonal.

Determine the point on yz-plane which is equidistant from points A(2, 0, 3), B(0, 3, 2) and C(0, 0, 1).

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.

The ratio in which the line joining (2, 4, 5) and (3, 5, -9) is divided by the yz-plane is

The ratio in which the line joining the points (a, b, c) and (-1, -c, -b) is divided by the xy-plane is

If P (0, 1, 2), Q(4, -2, 1) and O(0, 0, 0) are three points, then ∠POQ =

If the extremities of the diagonal of a square are (1, -2, 3) and (2, -3, 5), then the length of the side is

The points (5, -4, 2), (4, -3, 1), (7, 6, 4) and (8, -7, 5) are the vertices of

In a three dimensional space the equation x^{2} – 5x + 6 = 0 represents

Let (3, 4, -1) and (-1, 2, 3) be the endpoints of a diameter of a sphere. Then, the radius of the sphere is equal to

XOZ-plane divides the join of (2, 3, 1) and (6, 7, 1)

What is the locus of a point for which y = 0, z = 0?

the coordinates of the foot of the perpendicular drawn from the point P(3, 4, 5) on the yz-plane are

The coordinates of the foot of the perpendicular from a point P(6, 7, 8) on the x-axis are

The perpendicular distance of the point P(6, 7, 8) from xy-plane is

The length of the perpendicular drawn from the point P(3, 4, 5) on the y-axis is

The perpendicular distance of the point P(3, 3, 4) from the x-axis is

The length of the perpendicular drawn from the point P(a, b, c) from z-axis is