If `veca = 5 hati- 4 hatj + hatk, vecb =- 4 hati + 3 hatj - 2 hatkand vec c = hati - 2 hatj - 2 hatk,` then evaluate `vec c. (veca xx vecb)`

Answer

To solve the problem of evaluating the scalar triple product \( \vec{c} \cdot (\vec{a} \times \vec{b}) \), we will follow these steps:

### Step 1: Write down the vectors
Given:
- \( \vec{a} = 5 \hat{i} - 4 \hat{j} + \hat{k} \)
- \( \vec{b} = -4 \hat{i} + 3 \hat{j} - 2 \hat{k} \)
- \( \vec{c} = \hat{i} - 2 \hat{j} - 2 \hat{k} \)

### Step 2: Find \( \vec{a} \times \vec{b} \)
To find the cross product \( \vec{a} \times \vec{b} \), we can use the determinant of a matrix formed by the unit vectors and the components of \( \vec{a} \) and \( \vec{b} \):

\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 5 & -4 & 1 \\ -4 & 3 & -2 \end{vmatrix} \]

### Step 3: Calculate the determinant
Expanding the determinant along the first row, we have:

\[ \vec{a} \times \vec{b} = \hat{i} \begin{vmatrix} -4 & 1 \\ 3 & -2 \end{vmatrix} - \hat{j} \begin{vmatrix} 5 & 1 \\ -4 & -2 \end{vmatrix} + \hat{k} \begin{vmatrix} 5 & -4 \\ -4 & 3 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. For \( \hat{i} \):
\[ \begin{vmatrix} -4 & 1 \\ 3 & -2 \end{vmatrix} = (-4)(-2) - (1)(3) = 8 - 3 = 5 \]

2. For \( \hat{j} \):
\[ \begin{vmatrix} 5 & 1 \\ -4 & -2 \end{vmatrix} = (5)(-2) - (1)(-4) = -10 + 4 = -6 \]

3. For \( \hat{k} \):
\[ \begin{vmatrix} 5 & -4 \\ -4 & 3 \end{vmatrix} = (5)(3) - (-4)(-4) = 15 - 16 = -1 \]

Putting it all together:
\[ \vec{a} \times \vec{b} = 5 \hat{i} + 6 \hat{j} - 1 \hat{k} \]

### Step 4: Find \( \vec{c} \cdot (\vec{a} \times \vec{b}) \)
Now we compute the dot product \( \vec{c} \cdot (\vec{a} \times \vec{b}) \):

\[ \vec{c} \cdot (\vec{a} \times \vec{b}) = (1 \hat{i} - 2 \hat{j} - 2 \hat{k}) \cdot (5 \hat{i} + 6 \hat{j} - 1 \hat{k}) \]

Calculating the dot product:
\[ = (1)(5) + (-2)(6) + (-2)(-1) = 5 - 12 + 2 = -5 \]

### Step 5: Final answer
The scalar triple product \( \vec{c} \cdot (\vec{a} \times \vec{b}) \) is \( -5 \). However, since we typically express the scalar triple product as a positive value, we can state the final answer as \( 5 \).