-12i+xj,3j-k,2i+j-15k find x if area of parallelopiped is 546

To find the value of \( x \) in the given vectors that form a parallelepiped with a specified volume, we need to use the formula for the volume of a parallelepiped defined by three vectors. The volume \( V \) can be calculated using the scalar triple product of the vectors. Let's break this down step by step.

Understanding the Vectors

The vectors provided are:

• Vector A: \( -12i + xj + 3k \)
• Vector B: \( 2i + j - 15k \)
• Vector C: \( 3j - k \)

Volume of the Parallelepiped

The volume \( V \) of a parallelepiped formed by vectors \( \mathbf{A} \), \( \mathbf{B} \), and \( \mathbf{C} \) is given by the absolute value of the scalar triple product:

V = | \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) |

Step 1: Calculate the Cross Product \( \mathbf{B} \times \mathbf{C} \)

First, we need to compute the cross product of vectors \( \mathbf{B} \) and \( \mathbf{C} \). The vectors are:

• Vector B: \( (2, 1, -15) \)
• Vector C: \( (0, 3, -1) \)

The cross product is calculated using the determinant of a matrix:

\[ \mathbf{B} \times \mathbf{C} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 1 & -15 \\ 0 & 3 & -1 \end{vmatrix} \]

Calculating this determinant gives:

• For \( \mathbf{i} \): \( 1 \cdot (-1) - (-15) \cdot 3 = -1 + 45 = 44 \)
• For \( \mathbf{j} \): \( - (2 \cdot (-1) - (-15) \cdot 0) = -(-2) = 2 \)
• For \( \mathbf{k} \): \( 2 \cdot 3 - 1 \cdot 0 = 6 \)

Thus, we have:

Vector \( \mathbf{B} \times \mathbf{C} = 44\mathbf{i} - 2\mathbf{j} + 6\mathbf{k}

Step 2: Calculate the Dot Product \( \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) \)

Now, we compute the dot product of vector \( \mathbf{A} \) with the result from the cross product:

Vector A: \( (-12, x, 3) \)

Dot product calculation:

\[ \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = (-12)(44) + (x)(-2) + (3)(6) \]

This simplifies to:

\[ -528 - 2x + 18 = -510 - 2x \]

Step 3: Set Up the Equation for Volume

We know the volume of the parallelepiped is 546, so we set up the equation:

\[ |-510 - 2x| = 546 \]

This absolute value equation gives us two cases to solve:

Case 1: \( -510 - 2x = 546 \)

Solving this, we get:

\[ -2x = 546 + 510 = 1056 \implies x = -528 \]

Case 2: \( -510 - 2x = -546 \)

Solving this case gives:

\[ -2x = -546 + 510 = -36 \implies x = 18 \]

Final Values for \( x \)

We have two potential solutions for \( x \): \( -528 \) and \( 18 \). Depending on the context of the problem, you may choose the value that fits best. If you need a positive value, then \( x = 18 \) would be the appropriate choice.