A bird moves from point `(1, -2)` to `(4, 2)` . If the speed of the bird is `10m//sec`, then find the velocity vector of the bird ?

Options

A. `5(hati-2hatj)`
B. `5(4hati+2hatj)`
C. `0.6hati+0.8hatj`
D. `6hati+8hatj`

Answer

To find the velocity vector of the bird moving from point \( (1, -2) \) to \( (4, 2) \) with a speed of \( 10 \, \text{m/s} \), we can follow these steps:

### Step 1: Identify the Points
Let point A be \( (1, -2) \) and point B be \( (4, 2) \).

### Step 2: Calculate the Displacement Vector
The displacement vector \( \vec{AB} \) can be calculated using the formula:
\[ \vec{AB} = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} \]
where \( (x_1, y_1) = (1, -2) \) and \( (x_2, y_2) = (4, 2) \).

Calculating the components:
- \( x_2 - x_1 = 4 - 1 = 3 \)
- \( y_2 - y_1 = 2 - (-2) = 2 + 2 = 4 \)

Thus, the displacement vector is:
\[ \vec{AB} = 3 \hat{i} + 4 \hat{j} \]

### Step 3: Calculate the Magnitude of the Displacement Vector
The magnitude of the displacement vector \( |\vec{AB}| \) is given by:
\[ |\vec{AB}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Calculate the Unit Vector in the Direction of Displacement
The unit vector \( \hat{AB} \) in the direction of the displacement vector is:
\[ \hat{AB} = \frac{\vec{AB}}{|\vec{AB}|} = \frac{3 \hat{i} + 4 \hat{j}}{5} = \frac{3}{5} \hat{i} + \frac{4}{5} \hat{j} \]

### Step 5: Calculate the Velocity Vector
The velocity vector \( \vec{V} \) is given by the product of the speed and the unit vector:
\[ \vec{V} = \text{speed} \times \hat{AB} = 10 \times \left( \frac{3}{5} \hat{i} + \frac{4}{5} \hat{j} \right) \]
Calculating this gives:
\[ \vec{V} = 10 \times \frac{3}{5} \hat{i} + 10 \times \frac{4}{5} \hat{j} = 6 \hat{i} + 8 \hat{j} \]

### Final Answer
Thus, the velocity vector of the bird is:
\[ \vec{V} = 6 \hat{i} + 8 \hat{j} \]

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