Ch. - 12 Introduction to Three Dimensional Geometry

Ch. - 12 Introduction to Three Dimensional Geometry

Chapter 12 Ex.12.1 Question 1

A point is on the \(x - \)axis. What are its \(y - \)coordinate and \(z - \)coordinates?

Solution

If a point is on the \(x - \)axis, then the coordinates of \(y\) and \(z\) are \(0.\)

So, the point is \(\left( x,0,0 \right)\).

Chapter 12 Ex.12.1 Question 2

A point is in the \(XZ - \)plane. What can you say about its \(y - \)coordinate?

Solution

If a point is in \(XZ - \)plane, then its \(y - \)coordinate is \(0.\)

Chapter 12 Ex.12.1 Question 3

Name the octants in which the following points lie:

\(\begin{align}&\left( {1,2,3} \right),\;\left( {4, - 2,3} \right),\;\left( {4, - 2, - 5} \right),\\&\left( {4,2, - 5} \right),\;\left( { - 4,2, - 5} \right),\;\left( { - 4,2,5} \right),\;\\&\left( { - 3, - 1,6} \right),\left( {2, - 4, - 7} \right). \end{align}\)

Solution

Here is the table which represents the octants:

Octants

I

II

III

IV

V

VI

VII

VIII

\(x\)

\( + \)

\( - \)

\( - \)

\( + \)

\( + \)

\( - \)

\( - \)

\( + \)

\(y\)

\( + \)

\( + \)

\( - \)

\( - \)

\( + \)

\( + \)

\( - \)

\( - \)

\(z\)

\( + \)

\( + \)

\( + \)

\( + \)

\( - \)

\( - \)

\( - \)

\( - \)

(i) \(\left( {1,2,3} \right)\)

Here \(x\) is positive, \(y\) is positive, and \(z\) is positive.

So, it lies in \(I\) octant.

(ii) \(\left( {4, - 2,3} \right)\)

Here \(x\) is positive, \(y\) is negative, and \(z\) is positive.

So, it lies in IV octant.

(iii) \(\left( {4, - 2, - 5} \right)\)

Here \(x\) is positive, \(y\) is negative, and \(z\) is negative.

So, it lies in VIII octant.

(iv) \(\left( {4,2, - 5} \right)\)

Here \(x\) is positive, \(y\) is positive, and \(z\) is negative.

So, it lies in V octant.

(v) \(\left( { - 4,2, - 5} \right)\)

Here \(x\) is negative, \(y\) is positive, and \(z\) is negative.

So, it lies in VI octant.

(vi) \(\left( { - 4,2,5} \right)\)

Here \(x\) is negative, \(y\) is positive, and \(z\) is positive.

So, it lies in II octant.

(vii) \(\left( { - 3, - 1,6} \right)\)

Here \(x\) is negative, \(y\) is negative, and \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) is positive.

So, it lies in III octant.

(viii) \(\left( {2, - 4, - 7} \right)\)

Here \(x\) is positive, \(y\) is negative, and \(z\) is negative.

So, it lies in VIII octant.

Chapter 12 Ex.12.1 Question 4

Fill in the blanks:

(i) The \(x\)-axis and \(y\)-axis taken together determine a plane known as _____________.

(ii) The coordinates of points in the \(\rm{XY}\)-plane are of the form _____________.

(iii) Coordinate planes divide the space into _____________octants..

Solution

(i) The x-axis and y-axis taken together determine a plane known as XY plane.

(ii) The coordinates of points in the \(\rm{XY}\)-plane are of the form (x, y, 0).

(iii) Coordinate planes divide the space into eight octants.

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