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

(i) \displaystyle (5,2, 3)       (ii) \displaystyle (-5, 4, 3)       (iii) \displaystyle (4, - 3, 5)        (iv) \displaystyle (7 , 4, - 3)        (v) \displaystyle (-5, -4,7)        (vi) \displaystyle (-5, - 3, -2)        (vii) \displaystyle (2, -5, -7)        (viii) \displaystyle (-7,2-5)

Answer:

(i) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (5,2, 3) are all positive. Therefore this point lies in \displaystyle XOYZ octant.

(ii) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (-5, 4, 3) are negative, positive and positive respectively.  Therefore this point lies in \displaystyle X'OYZ octant.

(iii) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (4, - 3, 5) are positive, negative and positive respectively. Therefore this point lies in \displaystyle XOY'Z octant.

(iv) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (7 , 4, - 3) are positive, positive and negative respectively. Therefore this point lies in \displaystyle XOYZ' octant.       

(v) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (-5, -4,7) are  negative, negative and positive respectively. Therefore this point lies in \displaystyle X'OY'Z octant.      

(vi) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (-5, - 3, -2) are all negative. Therefore this point lies in \displaystyle X'OY'Z' octant.      

(vii) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (2, -5, -7)   are positive, negative and negative respectively. Therefore this point lies in \displaystyle XOY'Z' octant.     

(viii) The x-coordinate, the y-coordinate and the z-coordinate of the point \displaystyle (-7,2-5) are negative, positive and negative respectively. Therefore this point lies in \displaystyle X'OYZ' octant.

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Question 2: Find the image of: 

(i) \displaystyle (-2, 3, 4) \text{ in the YZ-plane. }

(ii) \displaystyle (- 5, 4, - 3) \text{ in the XZ-plane.}

(iii) \displaystyle (5, 2, -7) \text{ in the XY-plane.}

(iv) \displaystyle (- 5, 0, 3) \text{ in the XZ- plane.}

(v) \displaystyle (- 4,0, 0) \text{ in the XY-plane.}

Answer:

(i) \displaystyle YZ plane is the x-axis, hence the sign of \displaystyle x will be changed. Hence the image is \displaystyle (2, 3, 4)

(ii) \displaystyle XZ plane is the y-axis, hence the sign of \displaystyle y will be changed. Hence the image is \displaystyle (-5, -4, -3)

(iii) \displaystyle XY plane is the z-axis, hence the sign of \displaystyle z will be changed. Hence the image is \displaystyle (5, 2, 7)

(iv) \displaystyle XZ plane is the y-axis, hence the sign of \displaystyle y will be changed. Hence the image is \displaystyle (-5, 0, 3)

(v) \displaystyle XY plane is the z-axis, hence the sign of \displaystyle z will be changed. Hence the image is \displaystyle (-4, 0, 0)

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Question 3: A cube of side 5 has one vertex at the point \displaystyle (1,0, -1), and the three edges from this vertex are, respectively, parallel to the negative \displaystyle x and \displaystyle y axes and positive \displaystyle z axis. Find the coordinates of the other vertices of the cube.

Answer:

Given: A cube has side 5 having one vertex at \displaystyle (1, 0, -1)

Side of cube \displaystyle = 5

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

So let the Point \displaystyle A(1, 0, -1) and \displaystyle AB, AD and \displaystyle AE is parallel to negative x-axis, negative y-axis and positive z-axis respectively.

Since side of cube \displaystyle = 5

Point B is \displaystyle (-4, 0, -1)

Point D is \displaystyle (1, -5, -1)

Point E is \displaystyle (1, 0, 4)

Now, EH is parallel to negative y-axis

Point H is \displaystyle (1, -5, 4)

HG is parallel to negative x-axis

Point G is \displaystyle (-4, -5, -1)

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

Point C is \displaystyle (-4, -5, -1)

Point F is \displaystyle (-4, 0, 4)

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Question 4: Planes are drawn parallel to the coordinate planes through the points \displaystyle (3,0, -1) and \displaystyle (-2,5,4). Find the lengths of the edges of the parallelopiped so formed.

Answer:

\displaystyle  P = ( 3, 0, -1) \text{ and } Q = ( -2, 5, 4)

284Clearly, \displaystyle PA, PB and \displaystyle PC are the lengths of the edges of the parallelopiped shown in the figure.

Clearly, \displaystyle PBEC, QDAF are planes parallel to yx-plane such that their distances from yz-plane are \displaystyle x_1 and \displaystyle x_2 respectively. So, 

\displaystyle  PA = ( \text{ distance between the planes } PBEC \text{ and } QDAF) \\ \\ =| x_2 - x_1 |= |-2-3| = |-5 |= 5

\displaystyle PB is the distance between the planes \displaystyle PAFC and \displaystyle BDQE which are parallel to zx-plane and are at a distance \displaystyle y_1 and \displaystyle y_2 , respectively, from zx-plane.

\displaystyle \therefore PB = |y_2 - y_1| = |5-0| = 5

Similarly, \displaystyle PC is the distance between the planes \displaystyle PBDA and \displaystyle CEQF which are at a distance \displaystyle z_1 and \displaystyle z_2 respectively, from xy-plane.

\displaystyle \therefore PC =| z_2- z_1 |= |4 - (-1)| = 5

Therefore the the lengths of the edges of the parallelopiped are \displaystyle 5, 5 and \displaystyle 5

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Question 5: Planes are drawn through the points \displaystyle (5,0,2) and \displaystyle (3, -2,5) parallel to the coordinate planes. Find the lengths of the edges of the rectangular parallelopiped so formed.284

Answer:

\displaystyle  P = ( 5, 0, 2) \text{ and } Q = ( 3, -2, 5)

Clearly, \displaystyle PA, PB and \displaystyle PC are the lengths of the edges of the parallelopiped shown in the figure.

Clearly, \displaystyle PBEC, QDAF are planes parallel to yx-plane such that their distances from yz-plane are \displaystyle x_1 and \displaystyle x_2 respectively. So, 

\displaystyle  PA = ( \text{ distance between the planes } PBEC \text{ and } QDAF) \\ \\ =| x_2 - x_1 |= |3-5| = |-2 |= 2

\displaystyle PB is the distance between the planes \displaystyle PAFC and \displaystyle BDQE which are parallel to zx-plane and are at a distance \displaystyle y_1 and \displaystyle y_2 , respectively, from zx-plane.

\displaystyle \therefore PB = |y_2 - y_1| = |-2-0| = 2

Similarly, \displaystyle PC is the distance between the planes \displaystyle PBDA and \displaystyle CEQF which are at a distance \displaystyle z_1 and \displaystyle z_2 respectively, from xy-plane.

\displaystyle \therefore PC =| z_2- z_1 |= |5-2| = 3

Therefore the the lengths of the edges of the parallelopiped are \displaystyle 5, 5 and \displaystyle 5

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Question 6: Find the distances of the point \displaystyle P (- 4,3,5) from the coordinate axes.

Answer:

Given: The point \displaystyle P (-4, 3, 5)

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

\displaystyle \text{Distance } = \sqrt{y^2 + z^2} = \sqrt{3^2 + 5^2} = \sqrt{9+25} = \sqrt{34}

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

\displaystyle \text{Distance } = \sqrt{x^2 + z^2} = \sqrt{(-4)^2 + 5^2} = \sqrt{16+25} = \sqrt{41}

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

\displaystyle \text{Distance } = \sqrt{x^2 + y^2} = \sqrt{(-4)z^2 + 5^2} = \sqrt{9+25} = \sqrt{34}

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Question 7: The coordinates of a point are \displaystyle ( 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.

Answer:

Given: 

Point \displaystyle (3, -2, 5)

The Absolute value of any point \displaystyle (x, y, z) is given by, \displaystyle \sqrt{x^2+y^2+z^2}

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

So let the point \displaystyle A(3, -2, 5)

Remaining 7 points are:

Point \displaystyle B(3, 2, 5) (By changing the sign of y coordinate)

Point \displaystyle C(-3, -2, 5) (By changing the sign of x coordinate)

Point \displaystyle D(3, -2, -5) (By changing the sign of z coordinate)

Point \displaystyle E(-3, 2, 5) (By changing the sign of x and y coordinate)

Point \displaystyle F(3, 2, -5) (By changing the sign of y and z coordinate)

Point \displaystyle G(-3, -2, -5) (By changing the sign of x and z coordinate)

Point \displaystyle H(-3, 2, -5) (By changing the sign of x, y and z coordinate)