P and Q are two vectors given as `P=(3hati+5hatj)` and `Q=(2hatj)`. Find the magnitude of the vector `(P-Q)`

Answer

To find the magnitude of the vector \( P - Q \), we will follow these steps:

### Step 1: Write down the vectors
Given:
\[ P = 3\hat{i} + 5\hat{j} \]
\[ Q = 2\hat{j} \]

### Step 2: Calculate \( P - Q \)
To find \( P - Q \), we subtract the components of vector \( Q \) from vector \( P \):
\[ P - Q = (3\hat{i} + 5\hat{j}) - (0\hat{i} + 2\hat{j}) \]
This simplifies to:
\[ P - Q = 3\hat{i} + (5 - 2)\hat{j} = 3\hat{i} + 3\hat{j} \]

### Step 3: Find the magnitude of \( P - Q \)
The magnitude of a vector \( A = a\hat{i} + b\hat{j} \) is given by:
\[ |A| = \sqrt{a^2 + b^2} \]
For our vector \( P - Q = 3\hat{i} + 3\hat{j} \):
\[ |P - Q| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} \]

### Step 4: Simplify the magnitude
We can simplify \( \sqrt{18} \):
\[ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]

### Final Answer
Thus, the magnitude of the vector \( P - Q \) is:
\[ |P - Q| = 3\sqrt{2} \]

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