A line AB in three-dimensional space makes angles `45^@` and `120^@` with the positive x-axis and the positive y-axis respectively. If AB makes an acute angle with the positive z-axis, then `theta` equals

Options

A. `45^@`
B. `60^@`
C. `75^@`
D. `30^@`

Answer

To solve the problem step by step, we need to find the angle \( \theta \) that line AB makes with the positive z-axis, given that it makes angles of \( 45^\circ \) with the positive x-axis and \( 120^\circ \) with the positive y-axis.

### Step 1: Identify the direction cosines
The direction cosines of a line are given by:
- \( \cos \alpha \) for the angle \( \alpha \) with the x-axis,
- \( \cos \beta \) for the angle \( \beta \) with the y-axis,
- \( \cos \gamma \) for the angle \( \gamma \) with the z-axis.

Given:
- \( \alpha = 45^\circ \)
- \( \beta = 120^\circ \)
- \( \gamma = \theta \)

### Step 2: Calculate the direction cosines
Using the cosine values:
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2} \)

Thus, we have:
- \( \cos \alpha = \frac{1}{\sqrt{2}} \)
- \( \cos \beta = -\frac{1}{2} \)
- \( \cos \gamma = \cos \theta \)

### Step 3: Use the property of direction cosines
The sum of the squares of the direction cosines is equal to 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

Substituting the values we have:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{2}\right)^2 + \cos^2 \theta = 1 \]

### Step 4: Simplify the equation
Calculating the squares:
\[ \frac{1}{2} + \frac{1}{4} + \cos^2 \theta = 1 \]

### Step 5: Combine the fractions
Finding a common denominator (which is 4):
\[ \frac{2}{4} + \frac{1}{4} + \cos^2 \theta = 1 \]
\[ \frac{3}{4} + \cos^2 \theta = 1 \]

### Step 6: Isolate \( \cos^2 \theta \)
Subtract \( \frac{3}{4} \) from both sides:
\[ \cos^2 \theta = 1 - \frac{3}{4} \]
\[ \cos^2 \theta = \frac{1}{4} \]

### Step 7: Solve for \( \cos \theta \)
Taking the square root:
\[ \cos \theta = \frac{1}{2} \]

### Step 8: Find the angle \( \theta \)
The angle \( \theta \) corresponding to \( \cos \theta = \frac{1}{2} \) is:
\[ \theta = 60^\circ \]

### Conclusion
Thus, the angle \( \theta \) that line AB makes with the positive z-axis is:
\[ \theta = 60^\circ \]