Question

If the volume of the tetrahedron, whose vertices are \( A(1, 2, 3), B(-3, -1, 1), C(2, 1, 3) \) and \( D(-1, 2, x) \), is \( \frac{11}{6} \) cubic units, then the value of \( x \) is:

• A. 3
• B. -2
• C. 4
• D. -1

Hint:

When calculating the volume of a tetrahedron using the determinant formula, ensure to carefully expand the matrix and solve for the unknown variable.

Answer 1

The Correct Option is A

The volume \( V \) of a tetrahedron formed by four vertices \( A(x_1, y_1, z_1), B(x_2, y_2, z_2), C(x_3, y_3, z_3), D(x_4, y_4, z_4) \) is given by the formula: \[ V = \frac{1}{6} \left| \text{det} \begin{bmatrix} x_1 & y_1 & z_1 & 1
x_2 & y_2 & z_2 & 1
x_3 & y_3 & z_3 & 1
x_4 & y_4 & z_4 & 1
\end{bmatrix} \right| \] Substituting the coordinates of the points \( A(1, 2, 3), B(-3, -1, 1), C(2, 1, 3), D(-1, 2, x) \) into the above matrix: \[ V = \frac{1}{6} \left| \text{det} \begin{bmatrix} 1 & 2 & 3 & 1
-3 & -1 & 1 & 1
2 & 1 & 3 & 1
-1 & 2 & x & 1
\end{bmatrix} \right| = \frac{11}{6} \] Now, calculate the determinant of the matrix and solve for \( x \). Upon solving, we find that \( x = 3 \).