Find magnitude of A-2 B +3 C, where, `A = 2hati+3hatj, B = hati + hatj and c =hatk.`

Explanation

To solve the problem of finding the magnitude of the vector expression \( A - 2B + 3C \), where \( A = 2\hat{i} + 3\hat{j} \), \( B = \hat{i} + \hat{j} \), and \( C = \hat{k} \), we can follow these steps:

### Step 1: Write down the vectors
We have:
- \( A = 2\hat{i} + 3\hat{j} \)
- \( B = \hat{i} + \hat{j} \)
- \( C = \hat{k} \)

### Step 2: Substitute the vectors into the expression
We need to calculate \( A - 2B + 3C \):
\[ A - 2B + 3C = (2\hat{i} + 3\hat{j}) - 2(\hat{i} + \hat{j}) + 3\hat{k} \]

### Step 3: Distribute the coefficients
Now, distribute the coefficients in the expression:
\[ = (2\hat{i} + 3\hat{j}) - (2\hat{i} + 2\hat{j}) + 3\hat{k} \]

### Step 4: Combine like terms
Now, combine the like terms:
\[ = (2\hat{i} - 2\hat{i}) + (3\hat{j} - 2\hat{j}) + 3\hat{k} \]
\[ = 0\hat{i} + 1\hat{j} + 3\hat{k} \]
\[ = \hat{j} + 3\hat{k} \]

### Step 5: Write the resultant vector
The resultant vector is:
\[ \hat{j} + 3\hat{k} \]

### Step 6: Find the magnitude of the resultant vector
The magnitude of a vector \( \mathbf{v} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
In our case, \( a = 0 \), \( b = 1 \), and \( c = 3 \):
\[ |\hat{j} + 3\hat{k}| = \sqrt{0^2 + 1^2 + 3^2} = \sqrt{0 + 1 + 9} = \sqrt{10} \]

### Final Answer
The magnitude of \( A - 2B + 3C \) is \( \sqrt{10} \).

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