A vector is represented by `3hati+hatj+2hatk`. Its length in `XY` plane is

Options

A. 2
B. `sqrt(14)`
C. `sqrt(10)`
D. `sqrt(5)`

Explanation

To find the length of the vector \( \vec{A} = 3\hat{i} + \hat{j} + 2\hat{k} \) in the XY plane, we will follow these steps:

### Step 1: Identify the components of the vector
The given vector has three components:
- \( \hat{i} \) component: 3
- \( \hat{j} \) component: 1
- \( \hat{k} \) component: 2

### Step 2: Exclude the k component
Since we are interested in the length of the vector in the XY plane, we will only consider the \( \hat{i} \) and \( \hat{j} \) components. Therefore, we will ignore the \( \hat{k} \) component (which is 2).

### Step 3: Calculate the length in the XY plane
The length (magnitude) of a vector in the XY plane can be calculated using the formula:
\[ |\vec{A}_{XY}| = \sqrt{(A_x)^2 + (A_y)^2} \]
where \( A_x \) is the \( \hat{i} \) component and \( A_y \) is the \( \hat{j} \) component.

Substituting the values:
\[ |\vec{A}_{XY}| = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10} \]

### Step 4: Conclusion
The length of the vector in the XY plane is \( \sqrt{10} \).

### Final Answer
The length of the vector in the XY plane is \( \sqrt{10} \).

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