Which of the following angles made by a line with coordinate axes are not possible ?

Question

Accepted Answer

To determine which set of angles made by a line with the coordinate axes is not possible, we need to consider the geometric properties of angles between a line and the coordinate axes in three-dimensional space.

### Explanation for Option 3: $30^{\circ}, 45^{\circ}, 60^{\circ}$

In three-dimensional space, a line can make angles $\alpha$, $\beta$, and $\gamma$ with the x-axis, y-axis, and z-axis, respectively. These angles must satisfy the following condition:

$$
\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) = 1
$$

For Option 3, let's calculate:

- $\cos(30^{\circ}) = \sqrt{3}/2$
- $\cos(45^{\circ}) = \sqrt{2}/2$
- $\cos(60^{\circ}) = 1/2$

Now, compute the sum of the squares of these cosines:

$$
\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 = \frac{3}{4} + \frac{2}{4} + \frac{1}{4} = \frac{6}{4} = \frac{3}{2}
$$

Since $\frac{3}{2} \neq 1$, the condition is not satisfied, making this set of angles impossible for a line in three-dimensional space.

### Explanation for Other Options:

- **Option 1: $30^{\circ}, 60^{\circ}, 90^{\circ}$**

- $\cos(30^{\circ}) = \sqrt{3}/2$
  - $\cos(60^{\circ}) = 1/2$
  - $\cos(90^{\circ}) = 0$

Compute the sum:

$$
  \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + 0^2 = \frac{3}{4} + \frac{1}{4} + 0 = 1
  $$

This satisfies the condition, so this set is possible.

- **Option 2: $45^{\circ}, 90^{\circ}, 45^{\circ}$**

- $\cos(45^{\circ}) = \sqrt{2}/2$
  - $\cos(90^{\circ}) = 0$

Compute the sum:

$$
  2 \times \left(\frac{\sqrt{2}}{2}\right)^2 + 0^2 = 2 \times \frac{2}{4} = 1
  $$

This satisfies the condition, so this set is possible.

- **Option 4: $60^{\circ}, 45^{\circ}, 60^{\circ}$**

- $\cos(60^{\circ}) = 1/2$
  - $\cos(45^{\circ}) = \sqrt{2}/2$

Compute the sum:

$$
  2 \times \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2 = 2 \times \frac{1}{4} + \frac{2}{4} = \frac{2}{4} + \frac{2}{4} = 1
  $$

This satisfies the condition, so this set is possible.

Thus, Option 3 is the only set of angles that does not satisfy the condition for angles made by a line with the coordinate axes, confirming it as the correct answer.