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 find the position vector of the particle after 3 seconds based on the given information.

### Step 1: Identify the initial position vector
The initial position of the particle is given as the point (3, -7). We can express this as a position vector:
\[ \mathbf{u_1} = 3 \hat{i} - 7 \hat{j} \]

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

### Step 2: Determine the direction vector and its magnitude
The direction of the particle's motion is given by the vector \(7 \hat{i} - 24 \hat{j}\). To find the unit vector in this direction, we first calculate the magnitude of the direction vector:
\[ \text{Magnitude} = \sqrt{(7^2) + (-24^2)} = \sqrt{49 + 576} = \sqrt{625} = 25 \]

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

### Step 3: Calculate the unit vector in the direction of motion
The unit vector \(\hat{v}\) in the direction of motion is obtained by dividing the direction vector by its magnitude:
\[ \hat{v} = \frac{7 \hat{i} - 24 \hat{j}}{25} = \frac{7}{25} \hat{i} - \frac{24}{25} \hat{j} \]

**Hint:** A unit vector has a magnitude of 1 and is obtained by dividing the vector by its magnitude.

### Step 4: Calculate the velocity vector
The speed of the particle is given as \(50 \, \text{m/s}\). The velocity vector \(\mathbf{v}\) can be calculated by multiplying the speed by the unit vector:
\[ \mathbf{v} = 50 \hat{v} = 50 \left(\frac{7}{25} \hat{i} - \frac{24}{25} \hat{j}\right) = 14 \hat{i} - 48 \hat{j} \]

**Hint:** The velocity vector is the product of speed and the unit direction vector.

### Step 5: Calculate the displacement after 3 seconds
The displacement \(\mathbf{d}\) after \(t = 3\) seconds can be calculated as:
\[ \mathbf{d} = \mathbf{v} \cdot t = (14 \hat{i} - 48 \hat{j}) \cdot 3 = 42 \hat{i} - 144 \hat{j} \]

**Hint:** Displacement can be found by multiplying the velocity vector by the time.

### Step 6: Calculate the final position vector
The final position vector \(\mathbf{u_2}\) after 3 seconds is given by the initial position vector plus the displacement:
\[ \mathbf{u_2} = \mathbf{u_1} + \mathbf{d} = (3 \hat{i} - 7 \hat{j}) + (42 \hat{i} - 144 \hat{j}) = (3 + 42) \hat{i} + (-7 - 144) \hat{j} = 45 \hat{i} - 151 \hat{j} \]

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

### Final Answer
The position vector of the particle after 3 seconds is:
\[ \mathbf{u_2} = 45 \hat{i} - 151 \hat{j} \]