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

Options

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

Answer

To solve the problem, we need to find the value of \((\vec{a} - \vec{b}) \cdot (3\vec{a} + \vec{b})\) given the vectors:

\[ \vec{a} = \hat{i} - 2\hat{j} + 5\hat{k} \]
\[ \vec{b} = 2\hat{i} + \hat{j} - 3\hat{k} \]

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

We start by subtracting vector \(\vec{b}\) from vector \(\vec{a}\):

\[ \vec{a} - \vec{b} = (\hat{i} - 2\hat{j} + 5\hat{k}) - (2\hat{i} + \hat{j} - 3\hat{k}) \]

Distributing the negative sign:

\[ = \hat{i} - 2\hat{j} + 5\hat{k} - 2\hat{i} - \hat{j} + 3\hat{k} \]

Now, combine the like terms:

\[ = (\hat{i} - 2\hat{i}) + (-2\hat{j} - \hat{j}) + (5\hat{k} + 3\hat{k}) \]
\[ = -\hat{i} - 3\hat{j} + 8\hat{k} \]

### Step 2: Calculate \(3\vec{a} + \vec{b}\)

Next, we calculate \(3\vec{a} + \vec{b}\):

\[ 3\vec{a} = 3(\hat{i} - 2\hat{j} + 5\hat{k}) = 3\hat{i} - 6\hat{j} + 15\hat{k} \]

Now add \(\vec{b}\):

\[ 3\vec{a} + \vec{b} = (3\hat{i} - 6\hat{j} + 15\hat{k}) + (2\hat{i} + \hat{j} - 3\hat{k}) \]

Combining the like terms:

\[ = (3\hat{i} + 2\hat{i}) + (-6\hat{j} + \hat{j}) + (15\hat{k} - 3\hat{k}) \]
\[ = 5\hat{i} - 5\hat{j} + 12\hat{k} \]

### Step 3: Calculate the dot product \((\vec{a} - \vec{b}) \cdot (3\vec{a} + \vec{b})\)

Now we need to find the dot product:

\[ (\vec{a} - \vec{b}) \cdot (3\vec{a} + \vec{b}) = (-\hat{i} - 3\hat{j} + 8\hat{k}) \cdot (5\hat{i} - 5\hat{j} + 12\hat{k}) \]

Calculating the dot product:

\[ = (-1)(5) + (-3)(-5) + (8)(12) \]
\[ = -5 + 15 + 96 \]
\[ = 106 \]

### Final Answer

Thus, the value of \((\vec{a} - \vec{b}) \cdot (3\vec{a} + \vec{b})\) is:

\[ \boxed{106} \]