A vector of magnitude 10 has its rectangular components as 8 and 6 along x and y axes. Find the angles it make with these axes.

Answer

To find the angles that a vector makes with the x-axis and y-axis given its rectangular components, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components of the Vector**:
The vector has components along the x-axis and y-axis:
- \( V_x = 8 \) (component along x-axis)
- \( V_y = 6 \) (component along y-axis)

2. **Calculate the Magnitude of the Vector**:
The magnitude of the vector \( V \) is given as 10 units. We can verify this using the Pythagorean theorem:
\[ V = \sqrt{V_x^2 + V_y^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
This confirms that the components are correct.

3. **Find the Angle with the x-axis**:
The angle \( \theta \) that the vector makes with the x-axis can be found using the cosine function:
\[ \cos(\theta) = \frac{V_x}{V} = \frac{8}{10} = \frac{4}{5} \]
To find \( \theta \), take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{4}{5}\right) \]
Using a calculator or trigonometric tables, we find:
\[ \theta \approx 37^\circ \]

4. **Find the Angle with the y-axis**:
The angle \( \phi \) that the vector makes with the y-axis can be calculated as:
\[ \phi = 90^\circ - \theta \]
Substituting the value of \( \theta \):
\[ \phi = 90^\circ - 37^\circ = 53^\circ \]

5. **Final Results**:
- The angle with the x-axis is \( 37^\circ \).
- The angle with the y-axis is \( 53^\circ \).

### Summary:
- Angle with x-axis: \( 37^\circ \)
- Angle with y-axis: \( 53^\circ \)