A vector makes angle with X,Y and Z axer that are in the ratio `1:2:1` respectively. The angle made by the vector with Y-axis is

Answer

To solve the problem, we need to find the angle made by a vector with the Y-axis given that the angles with the X, Y, and Z axes are in the ratio 1:2:1. Let's denote the angles made with the X, Y, and Z axes as α, β, and γ, respectively.

### Step-by-Step Solution:

1. **Define the Angles**:
Since the angles are in the ratio 1:2:1, we can express them in terms of a variable, say β (the angle with the Y-axis):
- α = β/2 (angle with X-axis)
- β = β (angle with Y-axis)
- γ = β/2 (angle with Z-axis)

2. **Use the Direction Cosine Condition**:
The direction cosines of a vector are given by cos(α), cos(β), and cos(γ). The sum of the squares of the direction cosines must equal 1:
\[ \cos^2(α) + \cos^2(β) + \cos^2(γ) = 1 \]

3. **Substitute the Angles**:
Substitute α, β, and γ in terms of β into the equation:
\[ \cos^2(β/2) + \cos^2(β) + \cos^2(β/2) = 1 \]
This simplifies to:
\[ 2\cos^2(β/2) + \cos^2(β) = 1 \]

4. **Solve for cos(β/2)**:
Rearranging gives:
\[ 2\cos^2(β/2) = 1 - \cos^2(β) \]
We know that \( \cos^2(β) = 1 - \sin^2(β) \), so we can express it as:
\[ 2\cos^2(β/2) = \sin^2(β) \]

5. **Use the Double Angle Identity**:
We can use the double angle identity for sine:
\[ \sin^2(β) = 4\sin^2(β/2)\cos^2(β/2) \]
This gives us:
\[ 2\cos^2(β/2) = 4\sin^2(β/2)\cos^2(β/2) \]
Dividing both sides by cos^2(β/2) (assuming it's not zero):
\[ 2 = 4\sin^2(β/2) \]
Thus:
\[ \sin^2(β/2) = \frac{1}{2} \]

6. **Find β/2**:
Taking the square root gives:
\[ \sin(β/2) = \frac{1}{\sqrt{2}} \implies β/2 = \frac{\pi}{4} \]

7. **Find β**:
Therefore:
\[ β = 2 \times \frac{\pi}{4} = \frac{\pi}{2} \]

### Conclusion:
The angle made by the vector with the Y-axis is \( \frac{\pi}{2} \) radians or 90 degrees.