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 of these

Explanation

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

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

### Step 2: Calculate the magnitude of the component in the \(x-y\) plane
The magnitude of the component in the \(x-y\) plane can be calculated using the formula:
\[ \text{Magnitude in } x-y \text{ plane} = \sqrt{A_x^2 + A_y^2} \]
Substituting the values:
\[ \text{Magnitude in } x-y \text{ plane} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Calculate the magnitude of the component along the \(z\)-axis
The magnitude of the component along the \(z\)-axis is simply the absolute value of \(A_z\):
\[ \text{Magnitude along } z \text{-axis} = |A_z| = |10| = 10 \]

### Step 4: Calculate the ratio of the magnitudes
Now, we can find the ratio of the magnitude of the component in the \(x-y\) plane to the magnitude of the component along the \(z\)-axis:
\[ \text{Ratio} = \frac{\text{Magnitude in } x-y \text{ plane}}{\text{Magnitude along } z \text{-axis}} = \frac{5}{10} = \frac{1}{2} \]

### Final Answer
Thus, the ratio of the magnitude of the component on the \(x-y\) plane to the component on the \(z\)-axis is \(\frac{1}{2}\).

---