Find the velocity of a particle A w.r.t. B if velocity of particle A due west is `5kmh^(-1)` and that of B due south is `3kmh^(-1)`.

Answer

To find the velocity of particle A with respect to particle B, we can follow these steps:

### Step 1: Define the velocities of particles A and B
- The velocity of particle A is given as 5 km/h due west. In vector form, this can be represented as:
\[ \vec{v_A} = -5 \hat{i} \text{ km/h} \]
(where \(\hat{i}\) represents the unit vector in the east-west direction, and negative indicates west).

- The velocity of particle B is given as 3 km/h due south. In vector form, this can be represented as:
\[ \vec{v_B} = -3 \hat{j} \text{ km/h} \]
(where \(\hat{j}\) represents the unit vector in the north-south direction, and negative indicates south).

### Step 2: Calculate the relative velocity of A with respect to B
The relative velocity of A with respect to B is given by the formula:
\[ \vec{v_{A/B}} = \vec{v_A} - \vec{v_B} \]
Substituting the values we have:
\[ \vec{v_{A/B}} = (-5 \hat{i}) - (-3 \hat{j}) = -5 \hat{i} + 3 \hat{j} \]

### Step 3: Write the resulting velocity vector
The resulting velocity vector of A with respect to B is:
\[ \vec{v_{A/B}} = -5 \hat{i} + 3 \hat{j} \text{ km/h} \]

### Step 4: Calculate the magnitude of the relative velocity
To find the magnitude of the relative velocity, we use the formula:
\[ |\vec{v_{A/B}}| = \sqrt{(-5)^2 + (3)^2} \]
Calculating this gives:
\[ |\vec{v_{A/B}}| = \sqrt{25 + 9} = \sqrt{34} \]
Calculating \(\sqrt{34}\) gives approximately:
\[ |\vec{v_{A/B}}| \approx 5.83 \text{ km/h} \]

### Final Answer
The velocity of particle A with respect to particle B is approximately \(5.83 \text{ km/h}\).

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