Given `vec P =2 hati -3 hatj +4 hatk and vec Q= hatj-2 hatk`. The magnitude of their resultant is

Options

A. `sqrt(3)`
B. `2sqrt(3)`
C. `3sqrt(3)`
D. `4sqrt(3)`

Answer

To find the magnitude of the resultant vector \(\vec{R}\) formed by the vectors \(\vec{P}\) and \(\vec{Q}\), we will follow these steps:

### Step 1: Write down the given vectors
We have:
\[ \vec{P} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \]
\[ \vec{Q} = 0 \hat{i} + 1 \hat{j} - 2 \hat{k} \]

### Step 2: Add the vectors to find the resultant vector \(\vec{R}\)
To find the resultant vector \(\vec{R} = \vec{P} + \vec{Q}\), we add the corresponding components of \(\vec{P}\) and \(\vec{Q}\):
- The \(i\) component: \(2 + 0 = 2\)
- The \(j\) component: \(-3 + 1 = -2\)
- The \(k\) component: \(4 - 2 = 2\)

Thus, the resultant vector \(\vec{R}\) is:
\[ \vec{R} = 2 \hat{i} - 2 \hat{j} + 2 \hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector \(\vec{R}\)
The magnitude of a vector \(\vec{R} = a \hat{i} + b \hat{j} + c \hat{k}\) is given by:
\[ |\vec{R}| = \sqrt{a^2 + b^2 + c^2} \]
For our resultant vector \(\vec{R} = 2 \hat{i} - 2 \hat{j} + 2 \hat{k}\):
- \(a = 2\)
- \(b = -2\)
- \(c = 2\)

Now, substituting these values into the magnitude formula:
\[ |\vec{R}| = \sqrt{(2)^2 + (-2)^2 + (2)^2} \]
\[ |\vec{R}| = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} \]

### Final Answer
The magnitude of the resultant vector \(\vec{R}\) is:
\[ |\vec{R}| = 2\sqrt{3} \]