A particle travels with speed `50ms^(-1)` from the point `(3,-7)` in a direction `7hat(i)-24(j)`. Find its position vector after `3s`.

Explanation

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the initial position vector
The initial position of the particle is given as the point (3, -7). In vector notation, this can be represented as:
\[ \mathbf{r_0} = 3\hat{i} - 7\hat{j} \]

**Hint:** Remember that the position vector is represented in terms of unit vectors \(\hat{i}\) and \(\hat{j}\).

### Step 2: Determine the direction vector
The direction of the particle's movement is given as \(7\hat{i} - 24\hat{j}\).

**Hint:** The direction vector indicates the path along which the particle is moving.

### Step 3: Calculate the magnitude of the direction vector
To find the unit vector in the direction of motion, we first need to calculate the magnitude of the direction vector:
\[ |\mathbf{d}| = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \]

**Hint:** The magnitude of a vector can be found using the Pythagorean theorem.

### Step 4: Find the unit vector in the direction of motion
Now, we can find the unit vector \(\hat{u}\) in the direction of the particle's motion by dividing the direction vector by its magnitude:
\[ \hat{u} = \frac{7\hat{i} - 24\hat{j}}{25} = \frac{7}{25}\hat{i} - \frac{24}{25}\hat{j} \]

**Hint:** A unit vector has a magnitude of 1 and indicates direction only.

### Step 5: Calculate the velocity vector
The velocity vector \(\mathbf{v}\) can be calculated by multiplying the speed (50 m/s) by the unit vector:
\[ \mathbf{v} = 50 \hat{u} = 50 \left(\frac{7}{25}\hat{i} - \frac{24}{25}\hat{j}\right) = 14\hat{i} - 48\hat{j} \]

**Hint:** The velocity vector combines both speed and direction.

### Step 6: Calculate the displacement after 3 seconds
Displacement \(\mathbf{d}\) can be calculated using the formula:
\[ \mathbf{d} = \mathbf{v} \cdot t = (14\hat{i} - 48\hat{j}) \cdot 3 = 42\hat{i} - 144\hat{j} \]

**Hint:** Displacement is the product of velocity and time.

### Step 7: Find the final position vector
The final position vector \(\mathbf{r}\) can be found by adding the initial position vector and the displacement vector:
\[ \mathbf{r} = \mathbf{r_0} + \mathbf{d} = (3\hat{i} - 7\hat{j}) + (42\hat{i} - 144\hat{j}) = (3 + 42)\hat{i} + (-7 - 144)\hat{j} = 45\hat{i} - 151\hat{j} \]

**Hint:** The final position vector is the sum of the initial position and the displacement.

### Final Answer
The position vector after 3 seconds is:
\[ \mathbf{r} = 45\hat{i} - 151\hat{j} \]