A vector is represented by `3hat(i)+hat(j)+2hat(k)` Its length in XY plane is

Options

A. 2
B. `sqrt14`
C. `sqrt10`
D. `sqrt5`

Answer

To find the length of the vector in the XY plane, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Vector Components**:
The given vector is represented as:
\[ \vec{r} = 3\hat{i} + 1\hat{j} + 2\hat{k} \]
Here, \(3\hat{i}\) is the component along the x-axis, \(1\hat{j}\) is the component along the y-axis, and \(2\hat{k}\) is the component along the z-axis.

2. **Consider Only the XY Plane**:
To find the length of the vector in the XY plane, we only need the x and y components. The z component does not contribute to the length in the XY plane. Therefore, we can ignore the \(2\hat{k}\) part.

The components in the XY plane are:
\[ \vec{r}_{xy} = 3\hat{i} + 1\hat{j} \]

3. **Calculate the Magnitude of the Vector in the XY Plane**:
The magnitude of a vector \(\vec{r}_{xy} = a\hat{i} + b\hat{j}\) is given by the formula:
\[ |\vec{r}_{xy}| = \sqrt{a^2 + b^2} \]
Here, \(a = 3\) and \(b = 1\). Substituting these values, we get:
\[ |\vec{r}_{xy}| = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]

4. **Final Result**:
The length of the vector in the XY plane is:
\[ |\vec{r}_{xy}| = \sqrt{10} \text{ units} \]