If `vec(a)= hat(i) - 2hat(j) + 5hat(k) and vec(b) = 2hat(i) + hat(j) -3hat(k)`, then `(vec(b) - vec(a)).(3 vec(a) +vec(b))` equal to?

Options

A. 106
B. `-106`
C. 53
D. `-53`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Define the vectors
We are given two vectors:
\[ \vec{a} = \hat{i} - 2\hat{j} + 5\hat{k} \]
\[ \vec{b} = 2\hat{i} + \hat{j} - 3\hat{k} \]

### Step 2: Calculate \(\vec{b} - \vec{a}\)
To find \(\vec{b} - \vec{a}\), we subtract the components of \(\vec{a}\) from \(\vec{b}\):
\[ \vec{b} - \vec{a} = (2\hat{i} + \hat{j} - 3\hat{k}) - (\hat{i} - 2\hat{j} + 5\hat{k}) \]
Calculating each component:
- For \(\hat{i}\): \(2 - 1 = 1\)
- For \(\hat{j}\): \(1 - (-2) = 1 + 2 = 3\)
- For \(\hat{k}\): \(-3 - 5 = -8\)

Thus,
\[ \vec{b} - \vec{a} = \hat{i} + 3\hat{j} - 8\hat{k} \]

### Step 3: Calculate \(3\vec{a} + \vec{b}\)
Next, we calculate \(3\vec{a}\):
\[ 3\vec{a} = 3(\hat{i} - 2\hat{j} + 5\hat{k}) = 3\hat{i} - 6\hat{j} + 15\hat{k} \]
Now, we add \(\vec{b}\):
\[ 3\vec{a} + \vec{b} = (3\hat{i} - 6\hat{j} + 15\hat{k}) + (2\hat{i} + \hat{j} - 3\hat{k}) \]
Calculating each component:
- For \(\hat{i}\): \(3 + 2 = 5\)
- For \(\hat{j}\): \(-6 + 1 = -5\)
- For \(\hat{k}\): \(15 - 3 = 12\)

Thus,
\[ 3\vec{a} + \vec{b} = 5\hat{i} - 5\hat{j} + 12\hat{k} \]

### Step 4: Calculate the dot product \((\vec{b} - \vec{a}) \cdot (3\vec{a} + \vec{b})\)
Now we compute the dot product:
\[ (\hat{i} + 3\hat{j} - 8\hat{k}) \cdot (5\hat{i} - 5\hat{j} + 12\hat{k}) \]
Calculating the dot product:
- For \(\hat{i}\): \(1 \cdot 5 = 5\)
- For \(\hat{j}\): \(3 \cdot (-5) = -15\)
- For \(\hat{k}\): \(-8 \cdot 12 = -96\)

Adding these together:
\[ 5 - 15 - 96 = 5 - 15 - 96 = -106 \]

### Final Answer
Thus, the value of \((\vec{b} - \vec{a}) \cdot (3\vec{a} + \vec{b})\) is:
\[ \boxed{-106} \]