If `alpha = 2i + 3j - k, beta = -i + 2j-4k, gamma = i+j+k` then the value of `(alpha xx beta).(alpha xx gamma)` is equal to

Options

A. 60
B. 64
C. 74
D. `-74`

Answer

To find the value of \((\alpha \times \beta) \cdot (\alpha \times \gamma)\) where \(\alpha = 2i + 3j - k\), \(\beta = -i + 2j - 4k\), and \(\gamma = i + j + k\), we will follow these steps:

### Step 1: Calculate \(\alpha \times \beta\)

To compute the cross product \(\alpha \times \beta\), we can use the determinant of a matrix formed by the unit vectors \(i\), \(j\), \(k\) and the components of \(\alpha\) and \(\beta\):

\[ \alpha \times \beta = \begin{vmatrix} i & j & k \\ 2 & 3 & -1 \\ -1 & 2 & -4 \end{vmatrix} \]

Calculating this determinant:

\[ = i \begin{vmatrix} 3 & -1 \\ 2 & -4 \end{vmatrix} - j \begin{vmatrix} 2 & -1 \\ -1 & -4 \end{vmatrix} + k \begin{vmatrix} 2 & 3 \\ -1 & 2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 3 & -1 \\ 2 & -4 \end{vmatrix} = (3 \cdot -4) - (-1 \cdot 2) = -12 + 2 = -10 \)

2. \( \begin{vmatrix} 2 & -1 \\ -1 & -4 \end{vmatrix} = (2 \cdot -4) - (-1 \cdot -1) = -8 - 1 = -9 \)

3. \( \begin{vmatrix} 2 & 3 \\ -1 & 2 \end{vmatrix} = (2 \cdot 2) - (3 \cdot -1) = 4 + 3 = 7 \)

Putting it all together:

\[ \alpha \times \beta = -10i + 9j + 7k \]

### Step 2: Calculate \(\alpha \times \gamma\)

Now we compute \(\alpha \times \gamma\) using a similar determinant:

\[ \alpha \times \gamma = \begin{vmatrix} i & j & k \\ 2 & 3 & -1 \\ 1 & 1 & 1 \end{vmatrix} \]

Calculating this determinant:

\[ = i \begin{vmatrix} 3 & -1 \\ 1 & 1 \end{vmatrix} - j \begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix} + k \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 3 & -1 \\ 1 & 1 \end{vmatrix} = (3 \cdot 1) - (-1 \cdot 1) = 3 + 1 = 4 \)

2. \( \begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix} = (2 \cdot 1) - (-1 \cdot 1) = 2 + 1 = 3 \)

3. \( \begin{vmatrix} 2 & 3 \\ 1 & 1 \end{vmatrix} = (2 \cdot 1) - (3 \cdot 1) = 2 - 3 = -1 \)

Putting it all together:

\[ \alpha \times \gamma = 4i - 3j - k \]

### Step 3: Calculate \((\alpha \times \beta) \cdot (\alpha \times \gamma)\)

Now we find the dot product:

\[ (\alpha \times \beta) \cdot (\alpha \times \gamma) = (-10i + 9j + 7k) \cdot (4i - 3j - k) \]

Calculating the dot product:

\[ = (-10 \cdot 4) + (9 \cdot -3) + (7 \cdot -1) \]

Calculating each term:

1. \(-10 \cdot 4 = -40\)

2. \(9 \cdot -3 = -27\)

3. \(7 \cdot -1 = -7\)

Adding these together:

\[ -40 - 27 - 7 = -74 \]

### Final Answer

The value of \((\alpha \times \beta) \cdot (\alpha \times \gamma)\) is \(-74\).

---