A bird moves from point `(1 m, -2 m, 3 m)` to `(4 m, 2 m, 3 m)`. If the speed of the bird is `10 m//s`, then the velocity vector of the bird in `m//s` is:

Options

A. `5(hat(i)-2hat(j)+3hat(k))`
B. `5(4hat(i)+2hat(j)+3hat(k))`
C. `0.6 hat(i)+0.8 hat(j)`
D. `6hat(i)+8hat(j)`

Explanation

To find the velocity vector of the bird moving from point \( A(1, -2, 3) \) to point \( B(4, 2, 3) \) with a speed of \( 10 \, \text{m/s} \), we can follow these steps:

### Step 1: Determine the displacement vector
The displacement vector \( \vec{AB} \) from point \( A \) to point \( B \) can be calculated using the formula:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Where:
- \( \vec{A} = (1, -2, 3) \)
- \( \vec{B} = (4, 2, 3) \)

Calculating the components:
\[ \vec{AB} = (4 - 1, 2 - (-2), 3 - 3) = (3, 4, 0) \]

### Step 2: Write the displacement vector in unit vector form
The displacement vector can be expressed as:
\[ \vec{AB} = 3 \hat{i} + 4 \hat{j} + 0 \hat{k} \]

### Step 3: Calculate the magnitude of the displacement vector
The magnitude \( |\vec{AB}| \) of the displacement vector is given by:
\[ |\vec{AB}| = \sqrt{(3)^2 + (4)^2 + (0)^2} = \sqrt{9 + 16 + 0} = \sqrt{25} = 5 \]

### Step 4: Find the unit vector in the direction of the displacement
The unit vector \( \hat{u} \) in the direction of \( \vec{AB} \) is calculated as:
\[ \hat{u} = \frac{\vec{AB}}{|\vec{AB}|} = \frac{3 \hat{i} + 4 \hat{j} + 0 \hat{k}}{5} = \frac{3}{5} \hat{i} + \frac{4}{5} \hat{j} \]

### Step 5: Calculate the velocity vector
The velocity vector \( \vec{v} \) is given by the product of the speed and the unit vector:
\[ \vec{v} = \text{speed} \times \hat{u} = 10 \, \text{m/s} \times \left( \frac{3}{5} \hat{i} + \frac{4}{5} \hat{j} \right) \]
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:
\[ \vec{v} = 6 \hat{i} + 8 \hat{j} \, \text{m/s} \]

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