If `overline(a)=3hat(i)-hat(j)+4hat(k), overline(b)=2hat(i)+3hat(j)-2hat(k), overline(c)=-5hat(i)+2hat(j)+3hat(k)`, then `overline(a)*(overline(b)timesoverline(c))=`

Options

A. `100`
B. `101`
C. `111`
D. `109`

Answer

To solve the problem, we need to calculate the dot product of vector **a** with the cross product of vectors **b** and **c**. This can be represented mathematically as:

\[ \overline{a} \cdot (\overline{b} \times \overline{c}) \]

Given:
\[ \overline{a} = 3\hat{i} - \hat{j} + 4\hat{k} \]
\[ \overline{b} = 2\hat{i} + 3\hat{j} - 2\hat{k} \]
\[ \overline{c} = -5\hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 1: Calculate the cross product \(\overline{b} \times \overline{c}\)

To find \(\overline{b} \times \overline{c}\), we can use the determinant of a matrix formed by the unit vectors and the components of vectors **b** and **c**:

\[ \overline{b} \times \overline{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -2 \\ -5 & 2 & 3 \end{vmatrix} \]

Calculating this determinant:

\[ = \hat{i} \begin{vmatrix} 3 & -2 \\ 2 & 3 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & -2 \\ -5 & 3 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 3 \\ -5 & 2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. For \(\hat{i}\):
\[ \begin{vmatrix} 3 & -2 \\ 2 & 3 \end{vmatrix} = (3 \cdot 3) - (-2 \cdot 2) = 9 + 4 = 13 \]

2. For \(-\hat{j}\):
\[ \begin{vmatrix} 2 & -2 \\ -5 & 3 \end{vmatrix} = (2 \cdot 3) - (-2 \cdot -5) = 6 - 10 = -4 \]
Thus, \(-(-4) = 4\).

3. For \(\hat{k}\):
\[ \begin{vmatrix} 2 & 3 \\ -5 & 2 \end{vmatrix} = (2 \cdot 2) - (3 \cdot -5) = 4 + 15 = 19 \]

Putting it all together, we have:
\[ \overline{b} \times \overline{c} = 13\hat{i} + 4\hat{j} + 19\hat{k} \]

### Step 2: Calculate the dot product \(\overline{a} \cdot (\overline{b} \times \overline{c})\)

Now we calculate:
\[ \overline{a} \cdot (13\hat{i} + 4\hat{j} + 19\hat{k}) \]
\[ = (3\hat{i} - \hat{j} + 4\hat{k}) \cdot (13\hat{i} + 4\hat{j} + 19\hat{k}) \]
\[ = 3 \cdot 13 + (-1) \cdot 4 + 4 \cdot 19 \]
\[ = 39 - 4 + 76 \]
\[ = 39 + 76 - 4 = 111 \]

### Final Answer:
\[ \overline{a} \cdot (\overline{b} \times \overline{c}) = 111 \]