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 `theta` 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 will use the relationship between the angles a line makes with the coordinate axes and the cosine of those angles.

### Step 1: Define the angles
Let:
- \( \alpha = 45^\circ \) (angle with the x-axis)
- \( \beta = 120^\circ \) (angle with the y-axis)
- \( \theta \) = angle with the z-axis (which we need to find)

### Step 2: Use the cosine relationship
According to the property of angles in three-dimensional space, the sum of the squares of the cosines of the angles is equal to 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \theta = 1 \]

### Step 3: Calculate \( \cos \alpha \) and \( \cos \beta \)
Now, we calculate:
- \( \cos \alpha = \cos(45^\circ) = \frac{1}{\sqrt{2}} \)
- \( \cos \beta = \cos(120^\circ) = -\frac{1}{2} \)

### Step 4: Substitute into the equation
Substituting these values into the equation:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{2}\right)^2 + \cos^2 \theta = 1 \]
This simplifies to:
\[ \frac{1}{2} + \frac{1}{4} + \cos^2 \theta = 1 \]

### Step 5: Combine the fractions
Now, we need to combine the fractions:
\[ \frac{1}{2} = \frac{2}{4} \]
So,
\[ \frac{2}{4} + \frac{1}{4} + \cos^2 \theta = 1 \]
This gives:
\[ \frac{3}{4} + \cos^2 \theta = 1 \]

### Step 6: Solve for \( \cos^2 \theta \)
Now, isolate \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 7: Find \( \cos \theta \)
Taking the square root gives:
\[ \cos \theta = \frac{1}{2} \]
Since \( \theta \) is an acute angle, we take the positive root.

### Step 8: Determine \( \theta \)
Now, we find \( \theta \):
\[ \theta = \cos^{-1}\left(\frac{1}{2}\right) = 60^\circ \]

### Final Answer
Thus, the acute angle \( \theta \) with the positive z-axis is:
\[ \theta = 60^\circ \]

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