Question

Let $\text{P}(x, y, z)$ be a point in the first octant, whose projection in the xy-plane is the point $\text{Q}$. Let $\text{OP} = \gamma$; the angle between $\text{OQ}$ and the positive x-axis be $\theta$; and the angle between $\text{OP}$ and the positive z-axis be $\phi$, where $\text{O}$ is the origin. Then the distance of $\text{P}$ from the x-axis is:

• A. $\gamma \sqrt{1 - \sin^2 \phi \cos^2 \theta}$
• B. $\gamma \sqrt{1 + \cos^2 \theta \sin^2 \phi}$
• C. $\gamma \sqrt{1 - \sin^2 \theta \cos^2 \phi}$
• D. $\gamma \sqrt{1 + \cos^2 \phi \sin^2 \theta}$

Answer 1

The Correct Option is A

To solve this problem, we need to understand the relationship between the point \(\text{P}(x, y, z)\), its projection \(\text{Q}\) on the xy-plane, and the angles involved.

We are given:

• \(\text{OP} = \gamma\) (the distance from the origin to the point \(\text{P}\)).
• \(\theta\) is the angle between \(\text{OQ}\) (projection on the xy-plane) and the positive x-axis.
• \(\phi\) is the angle between \(\text{OP}\) and the positive z-axis.

The task is to find the distance of the point \(\text{P}\) from the x-axis. This distance can be represented as \(\sqrt{y^2 + z^2}\), where \(y\) and \(z\) are the y and z coordinates of the point \(\text{P}\).

Using spherical coordinates representation:

• The z coordinate is given by \(z = \gamma \cos \phi\).
• The xy-projection distance (i.e., \(|\text{OQ}|\text{) is given by\)
• The x coordinate in terms of this projection is \(x = \gamma \sin \phi \cos \theta\).
• The y coordinate is then: \(y = \gamma \sin \phi \sin \theta\).

The distance from the x-axis, \(\sqrt{y^2 + z^2}\), is calculated step-by-step as follows:

• Substitute \(y\) and \(z\) into the distance formula:

\(\sqrt{(\gamma \sin \phi \sin \theta)^2 + (\gamma \cos \phi)^2} = \sqrt{\gamma^2 \sin^2 \phi \sin^2 \theta + \gamma^2 \cos^2 \phi}\).

• Factor out \(\gamma^2\):

\(\gamma \sqrt{\sin^2 \phi \sin^2 \theta + \cos^2 \phi}\).

• Use the identity \(\sin^2 \theta + \cos^2 \theta = 1\):

\(\gamma \sqrt{1 - \cos^2 \theta \sin^2 \phi}\).

Thus, the distance of point \(\text{P}\) from the x-axis is: \(\gamma \sqrt{1 - \sin^2 \phi \cos^2 \theta}\).

Hence, the correct answer is \(\gamma \sqrt{1 - \sin^2 \phi \cos^2 \theta}\).

Answer 2

To find the distance of the point \(P(x, y, z)\) from the x-axis, we first need to understand the spatial configuration described in the question.

Given:

• \(OP = \gamma\): The distance from the origin \(O\) to the point \(P\).
• \(\theta\): The angle between the projection \(OQ\) on the xy-plane and the positive x-axis.
• \(\phi\): The angle between \(OP\) and the positive z-axis.

We need to find the distance from \(P\) to the x-axis. This distance can be represented in terms of the y and z coordinates of point \(P\) since any point on the x-axis will have z = 0 and y = 0.

Steps to solve:

1. From the given information, the coordinates of \(P\) in terms of spherical coordinates will be converted to Cartesian coordinates:
   • \(x = \gamma \cos\theta \sin\phi\)
   • \(y = \gamma \sin\theta \sin\phi\)
   • \(z = \gamma \cos\phi\)
2. The distance \(d\) from the x-axis, using the formula for the distance from a line, can be found as:
   • The distance \(d = \sqrt{y^2 + z^2}\)
   • Substitute the values of \(y\) and \(z\) from above:
   • \(d = \sqrt{(\gamma \sin \theta \sin \phi)^2 + (\gamma \cos \phi)^2}\)
   • \(d = \gamma \sqrt{\sin^2 \theta \sin^2 \phi + \cos^2 \phi}\)
3. Simplify the expression:
   • Using the identity \(\cos^2 \phi = 1 - \sin^2 \phi\), we can rewrite the equation:
   • \(d = \gamma \sqrt{\sin^2 \phi (\sin^2 \theta + \cos^2 \phi/\sin^2\phi)}\)
   • Simplify further to get:
4. \(d = \gamma \sqrt{1 - \sin^2 \phi \cos^2 \theta}\)

Thus, the distance of \(P\) from the x-axis is \(\gamma \sqrt{1 - \sin^2 \phi \cos^2 \theta}\), which matches with the correct answer given in the options.