The velociies of A and B are `vecv_(A)= 2hati + 4hatj and vecv_(B) = 3hati - 7hatj` , velocity of B as observed by A is

Options

A. `5hati-3hatj`
B. `hati - 11 hatj`
C. `-hati + 11hatj`
D. `2hati - 3hatj`

Answer

To find the velocity of B as observed by A, we can use the formula:

\[ \vec{v}_{BA} = \vec{v}_B - \vec{v}_A \]

Where:
- \(\vec{v}_{BA}\) is the velocity of B as observed by A.
- \(\vec{v}_B\) is the velocity of B.
- \(\vec{v}_A\) is the velocity of A.

Given:
- \(\vec{v}_A = 2\hat{i} + 4\hat{j}\)
- \(\vec{v}_B = 3\hat{i} - 7\hat{j}\)

### Step 1: Write down the velocities
We have:
\[ \vec{v}_A = 2\hat{i} + 4\hat{j} \]
\[ \vec{v}_B = 3\hat{i} - 7\hat{j} \]

### Step 2: Substitute the values into the formula
Now we substitute these values into the equation:
\[ \vec{v}_{BA} = (3\hat{i} - 7\hat{j}) - (2\hat{i} + 4\hat{j}) \]

### Step 3: Distribute the negative sign
Distributing the negative sign gives us:
\[ \vec{v}_{BA} = 3\hat{i} - 7\hat{j} - 2\hat{i} - 4\hat{j} \]

### Step 4: Combine like terms
Now, combine the \(\hat{i}\) and \(\hat{j}\) components:
\[ \vec{v}_{BA} = (3 - 2)\hat{i} + (-7 - 4)\hat{j} \]
\[ \vec{v}_{BA} = 1\hat{i} - 11\hat{j} \]

### Final Result
Thus, the velocity of B as observed by A is:
\[ \vec{v}_{BA} = \hat{i} - 11\hat{j} \]