Find the angles made by the line AB with the positive directions of the coordinate axes, if A is `(0,sqrt3,0)` and B is (0,0,-1).

Options

A. `(pi)/(2),(5pi)/(6),(2pi)/(3)`
B. `(pi)/(2),(pi)/(3),(pi)/(6)`
C. `(pi)/(2),(5pi)/(6),(pi)/(3)`
D. `(pi)/(3),(pi)/(6),(5pi)/(6)`

Answer

To find the angles made by the line AB with the positive directions of the coordinate axes, we will follow these steps:

### Step 1: Find the Direction Ratios of Line AB
The coordinates of points A and B are given as:
- A = (0, √3, 0)
- B = (0, 0, -1)

The direction ratios of the line AB can be calculated using the formula:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
Substituting the coordinates:
\[ \text{Direction Ratios} = (0 - 0, 0 - \sqrt{3}, -1 - 0) = (0, -\sqrt{3}, -1) \]

### Step 2: Calculate the Magnitude of the Direction Ratios
The magnitude of the direction ratios can be calculated as:
\[ \text{Magnitude} = \sqrt{(0)^2 + (-\sqrt{3})^2 + (-1)^2} = \sqrt{0 + 3 + 1} = \sqrt{4} = 2 \]

### Step 3: Find the Direction Cosines
The direction cosines (l, m, n) can be calculated by dividing each direction ratio by the magnitude:
- \( l = \frac{0}{2} = 0 \)
- \( m = \frac{-\sqrt{3}}{2} \)
- \( n = \frac{-1}{2} \)

### Step 4: Relate Direction Cosines to Angles
The direction cosines are related to the angles (α, β, γ) made with the positive x, y, and z axes respectively:
- \( \cos \alpha = l = 0 \)
- \( \cos \beta = m = -\frac{\sqrt{3}}{2} \)
- \( \cos \gamma = n = -\frac{1}{2} \)

### Step 5: Calculate the Angles
Using the inverse cosine function:
1. For \( \cos \alpha = 0 \):
- \( \alpha = \cos^{-1}(0) = 90^\circ \) (or \( \frac{\pi}{2} \) radians)

2. For \( \cos \beta = -\frac{\sqrt{3}}{2} \):
- \( \beta = \cos^{-1}\left(-\frac{\sqrt{3}}{2}\right) = 150^\circ \) (or \( \frac{5\pi}{6} \) radians)

3. For \( \cos \gamma = -\frac{1}{2} \):
- \( \gamma = \cos^{-1}\left(-\frac{1}{2}\right) = 120^\circ \) (or \( \frac{2\pi}{3} \) radians)

### Final Result
The angles made by the line AB with the positive directions of the coordinate axes are:
- \( \alpha = 90^\circ \)
- \( \beta = 150^\circ \)
- \( \gamma = 120^\circ \)