If `a=2hat(i)+3hat(j)-hat(k), b=-hat(i)+2hat(j)-4hat(k), c=hat(i)+hat(j)+hat(k)`, then find the value of `(atimesb)*(atimesc)`.

Answer

To solve the problem, we need to find the value of \((\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{c})\) given the vectors:

\[ \mathbf{a} = 2\hat{i} + 3\hat{j} - \hat{k} \]
\[ \mathbf{b} = -\hat{i} + 2\hat{j} - 4\hat{k} \]
\[ \mathbf{c} = \hat{i} + \hat{j} + \hat{k} \]

### Step 1: Calculate \(\mathbf{a} \times \mathbf{b}\)

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

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

Calculating this determinant:

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

Calculating the minors:

1. \(\begin{vmatrix} 3 & -1 \\ 2 & -4 \end{vmatrix} = (3)(-4) - (-1)(2) = -12 + 2 = -10\)
2. \(\begin{vmatrix} 2 & -1 \\ -1 & -4 \end{vmatrix} = (2)(-4) - (-1)(-1) = -8 - 1 = -9\)
3. \(\begin{vmatrix} 2 & 3 \\ -1 & 2 \end{vmatrix} = (2)(2) - (3)(-1) = 4 + 3 = 7\)

Putting it all together:

\[ \mathbf{a} \times \mathbf{b} = -10\hat{i} + 9\hat{j} + 7\hat{k} \]

### Step 2: Calculate \(\mathbf{a} \times \mathbf{c}\)

Now, we calculate \(\mathbf{a} \times \mathbf{c}\):

\[ \mathbf{a} \times \mathbf{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ 1 & 1 & 1 \end{vmatrix} \]

Calculating this determinant:

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

Calculating the minors:

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

Putting it all together:

\[ \mathbf{a} \times \mathbf{c} = 4\hat{i} - 3\hat{j} - 1\hat{k} \]

### Step 3: Calculate \((\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{c})\)

Now we need to find the dot product:

\[ (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{c}) = (-10\hat{i} + 9\hat{j} + 7\hat{k}) \cdot (4\hat{i} - 3\hat{j} - 1\hat{k}) \]

Calculating the dot product:

\[ = (-10)(4) + (9)(-3) + (7)(-1) \]
\[ = -40 - 27 - 7 \]
\[ = -74 \]

### Final Answer

Thus, the value of \((\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{a} \times \mathbf{c})\) is \(-74\).

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