If `veca = 3hati - hatj - 4hatk , vecb = -2hati + 4hatj - 3hatk` and `vec c = hati + 2hatj - hatk` then `|3veca - 2vec b + 4vec c |` is equal to

Options

A. `sqrt298`
B. `sqrt198`
C. `sqrt398`
D. `sqrt498`

Answer

To solve the problem, we need to calculate the expression \( |3\vec{a} - 2\vec{b} + 4\vec{c}| \) given the vectors:

\[ \vec{a} = 3\hat{i} - \hat{j} - 4\hat{k} \]
\[ \vec{b} = -2\hat{i} + 4\hat{j} - 3\hat{k} \]
\[ \vec{c} = \hat{i} + 2\hat{j} - \hat{k} \]

### Step 1: Calculate \( 3\vec{a} \)

\[ 3\vec{a} = 3(3\hat{i} - \hat{j} - 4\hat{k}) = 9\hat{i} - 3\hat{j} - 12\hat{k} \]

### Step 2: Calculate \( -2\vec{b} \)

\[ -2\vec{b} = -2(-2\hat{i} + 4\hat{j} - 3\hat{k}) = 4\hat{i} - 8\hat{j} + 6\hat{k} \]

### Step 3: Calculate \( 4\vec{c} \)

\[ 4\vec{c} = 4(\hat{i} + 2\hat{j} - \hat{k}) = 4\hat{i} + 8\hat{j} - 4\hat{k} \]

### Step 4: Combine the results

Now we combine \( 3\vec{a} \), \( -2\vec{b} \), and \( 4\vec{c} \):

\[ 3\vec{a} - 2\vec{b} + 4\vec{c} = (9\hat{i} - 3\hat{j} - 12\hat{k}) + (4\hat{i} - 8\hat{j} + 6\hat{k}) + (4\hat{i} + 8\hat{j} - 4\hat{k}) \]

Combining the coefficients of \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \):

- For \( \hat{i} \):
\[ 9 + 4 + 4 = 17 \]
- For \( \hat{j} \):
\[ -3 - 8 + 8 = -3 \]
- For \( \hat{k} \):
\[ -12 + 6 - 4 = -10 \]

Thus, we have:

\[ 3\vec{a} - 2\vec{b} + 4\vec{c} = 17\hat{i} - 3\hat{j} - 10\hat{k} \]

### Step 5: Calculate the magnitude

Now we calculate the magnitude:

\[ |3\vec{a} - 2\vec{b} + 4\vec{c}| = \sqrt{(17)^2 + (-3)^2 + (-10)^2} \]

Calculating each term:

- \( 17^2 = 289 \)
- \( (-3)^2 = 9 \)
- \( (-10)^2 = 100 \)

Adding these values:

\[ |3\vec{a} - 2\vec{b} + 4\vec{c}| = \sqrt{289 + 9 + 100} = \sqrt{398} \]

### Final Answer

Thus, the final result is:

\[ |3\vec{a} - 2\vec{b} + 4\vec{c}| = \sqrt{398} \]

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