Question 1: Name the octants in which the following points lie:

(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Answer:

(i) The x-coordinate, the y-coordinate and the z-coordinate of the point are all positive. Therefore this point lies in octant.

(ii) The x-coordinate, the y-coordinate and the z-coordinate of the point are negative, positive and positive respectively. Therefore this point lies in octant.

(iii) The x-coordinate, the y-coordinate and the z-coordinate of the point are positive, negative and positive respectively. Therefore this point lies in octant.

(iv) The x-coordinate, the y-coordinate and the z-coordinate of the point are positive, positive and negative respectively. Therefore this point lies in octant.

(v) The x-coordinate, the y-coordinate and the z-coordinate of the point are negative, negative and positive respectively. Therefore this point lies in octant.

(vi) The x-coordinate, the y-coordinate and the z-coordinate of the point are all negative. Therefore this point lies in octant.

(vii) The x-coordinate, the y-coordinate and the z-coordinate of the point are positive, negative and negative respectively. Therefore this point lies in octant.

(viii) The x-coordinate, the y-coordinate and the z-coordinate of the point are negative, positive and negative respectively. Therefore this point lies in octant.

Question 2: Find the image of:

(i)

(ii)

(iii)

(iv)

(v)

Answer:

(i) plane is the x-axis, hence the sign of will be changed. Hence the image is

(ii) plane is the y-axis, hence the sign of will be changed. Hence the image is

(iii) plane is the z-axis, hence the sign of will be changed. Hence the image is

(iv) plane is the y-axis, hence the sign of will be changed. Hence the image is

(v) plane is the z-axis, hence the sign of will be changed. Hence the image is

Question 3: A cube of side 5 has one vertex at the point and the three edges from this vertex are, respectively, parallel to the negative and axes and positive axis. Find the coordinates of the other vertices of the cube.

Answer:

Given: A cube has side 5 having one vertex at

Side of cube

We need to find the coordinates of the other vertices of the cube.

So let the Point and and is parallel to negative x-axis, negative y-axis and positive z-axis respectively.

Since side of cube

Point B is

Point D is

Point E is

Now, EH is parallel to negative y-axis

Point H is

HG is parallel to negative x-axis

Point G is

Now, again GC and GF is parallel to negative z-axis and positive y-axis respectively

Point C is

Point F is

Question 4: Planes are drawn parallel to the coordinate planes through the points and Find the lengths of the edges of the parallelopiped so formed.

Answer:

Clearly, and are the lengths of the edges of the parallelopiped shown in the figure.

Clearly, are planes parallel to yx-plane such that their distances from yz-plane are and respectively. So,

is the distance between the planes and which are parallel to zx-plane and are at a distance and , respectively, from zx-plane.

Similarly, is the distance between the planes and which are at a distance and respectively, from xy-plane.

Therefore the the lengths of the edges of the parallelopiped are and

Question 5: Planes are drawn through the points and parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelopiped so formed.

Answer:

Clearly, and are the lengths of the edges of the parallelopiped shown in the figure.

Clearly, are planes parallel to yx-plane such that their distances from yz-plane are and respectively. So,

is the distance between the planes and which are parallel to zx-plane and are at a distance and , respectively, from zx-plane.

Similarly, is the distance between the planes and which are at a distance and respectively, from xy-plane.

Therefore the the lengths of the edges of the parallelopiped are and

Question 6: Find the distances of the point from the coordinate axes.

Answer:

Given: The point

The distance of the point from x-axis is given as:

The distance of the point from y-axis is given as:

The distance of the point from z-axis is given as:

Question 7: The coordinates of a point are 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.

Answer:

Given:

Point

The Absolute value of any point is given by,

We need to make sure that absolute value to be the same for all points.

So let the point

Remaining 7 points are:

Point (By changing the sign of y coordinate)

Point (By changing the sign of x coordinate)

Point (By changing the sign of z coordinate)

Point (By changing the sign of x and y coordinate)

Point (By changing the sign of y and z coordinate)

Point (By changing the sign of x and z coordinate)

Point (By changing the sign of x, y and z coordinate)