Obtain the magnitude of 2A - 3B if
`A = hati +hatj- 2 hat k and B = 2 hat i - hat j+hat k` .

Options

A. √
150
B. 90
C. √
90
D. √
80

Explanation

To solve the problem of finding the magnitude of \( 2A - 3B \) given the vectors \( A \) and \( B \), we will follow these steps:

### Step 1: Write down the vectors
Given:
\[ A = \hat{i} + \hat{j} - 2\hat{k} \]
\[ B = 2\hat{i} - \hat{j} + \hat{k} \]

### Step 2: Calculate \( 2A \)
We need to calculate \( 2A \):
\[ 2A = 2(\hat{i} + \hat{j} - 2\hat{k}) = 2\hat{i} + 2\hat{j} - 4\hat{k} \]

### Step 3: Calculate \( 3B \)
Next, we calculate \( 3B \):
\[ 3B = 3(2\hat{i} - \hat{j} + \hat{k}) = 6\hat{i} - 3\hat{j} + 3\hat{k} \]

### Step 4: Compute \( 2A - 3B \)
Now we will compute \( 2A - 3B \):
\[ 2A - 3B = (2\hat{i} + 2\hat{j} - 4\hat{k}) - (6\hat{i} - 3\hat{j} + 3\hat{k}) \]
Distributing the negative sign:
\[ = 2\hat{i} + 2\hat{j} - 4\hat{k} - 6\hat{i} + 3\hat{j} - 3\hat{k} \]
Combining like terms:
\[ = (2 - 6)\hat{i} + (2 + 3)\hat{j} + (-4 - 3)\hat{k} \]
\[ = -4\hat{i} + 5\hat{j} - 7\hat{k} \]

### Step 5: Find the magnitude of \( 2A - 3B \)
The magnitude of a vector \( \vec{V} = x\hat{i} + y\hat{j} + z\hat{k} \) is given by:
\[ |\vec{V}| = \sqrt{x^2 + y^2 + z^2} \]
For our vector \( -4\hat{i} + 5\hat{j} - 7\hat{k} \):
\[ |\vec{V}| = \sqrt{(-4)^2 + (5)^2 + (-7)^2} \]
Calculating each term:
\[ = \sqrt{16 + 25 + 49} \]
\[ = \sqrt{90} \]

### Final Answer
Thus, the magnitude of \( 2A - 3B \) is:
\[ |\vec{V}| = \sqrt{90} \]

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