For the given vector `vec(A)=3hat(i)+4hat(j)+10hat(k)`, the ratio of magnitude of its component on the `x-y` plane and the component on `z-`axis is

Options

A. `2`
B. `(1)/(2)`
C. `1`
D. None

Answer

To solve the problem, we need to find the ratio of the magnitude of the components of the vector \(\vec{A} = 3\hat{i} + 4\hat{j} + 10\hat{k}\) on the \(x-y\) plane and the \(z\)-axis.

### Step-by-Step Solution:

1. **Identify the components of the vector:**
The vector \(\vec{A}\) has the following components:
- \(A_x = 3\) (coefficient of \(\hat{i}\))
- \(A_y = 4\) (coefficient of \(\hat{j}\))
- \(A_z = 10\) (coefficient of \(\hat{k}\))

2. **Calculate the magnitude of the component on the \(x-y\) plane:**
The component of the vector on the \(x-y\) plane is given by the vector formed by its \(x\) and \(y\) components:
\[ \vec{A}_{xy} = 3\hat{i} + 4\hat{j} \]
The magnitude of this vector can be calculated using the formula:
\[ |\vec{A}_{xy}| = \sqrt{A_x^2 + A_y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Calculate the magnitude of the component on the \(z\)-axis:**
The component of the vector on the \(z\)-axis is simply:
\[ A_z = 10 \]
Therefore, the magnitude of the component on the \(z\)-axis is:
\[ |\vec{A}_{z}| = 10 \]

4. **Calculate the ratio of the magnitudes:**
Now, we need to find the ratio of the magnitude of the component on the \(x-y\) plane to the magnitude of the component on the \(z\)-axis:
\[ \text{Ratio} = \frac{|\vec{A}_{xy}|}{|\vec{A}_{z}|} = \frac{5}{10} = \frac{1}{2} \]

5. **Final Answer:**
The ratio of the magnitude of the component of the vector on the \(x-y\) plane to the component on the \(z\)-axis is \(\frac{1}{2}\).