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

Options

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

Explanation

To solve the problem, we need to compute \((\vec{b} - \vec{a}) \cdot (3\vec{a} + \vec{b})\) where \(\vec{a} = \hat{i} - 2\hat{j} + 5\hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{j} - 3\hat{k}\).

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

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

Distributing the negative sign:

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

Combining like terms:

\[ \vec{b} - \vec{a} = (2 - 1)\hat{i} + (1 + 2)\hat{j} + (-3 - 5)\hat{k} = \hat{i} + 3\hat{j} - 8\hat{k} \]

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

First, 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 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 like terms:

\[ 3\vec{a} + \vec{b} = (3 + 2)\hat{i} + (-6 + 1)\hat{j} + (15 - 3)\hat{k} = 5\hat{i} - 5\hat{j} + 12\hat{k} \]

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

Now we compute:

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

Using the dot product formula:

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

Calculating each term:

\[ = 5 - 15 - 96 \]

Combining these results:

\[ = 5 - 15 - 96 = -10 - 96 = -106 \]

### Final Result:

Thus, the value of \((\vec{b} - \vec{a}) \cdot (3\vec{a} + \vec{b})\) is \(-106\).

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