Question

The volume of the tetrahedron with vertices \( P(1, 2, 0), Q(2, 1, -3), R(1, 0, 1) \), and \( S(3, -2, 3) \) is:

• A. \( 1 \)
• B. \( \frac{2}{3} \)
• C. \( 2 \)
• D. \( \frac{4}{3} \)

Hint:

Use the determinant formula to find the volume of a tetrahedron with given vertices.

Answer 1

The Correct Option is B

Step 1: Volume of tetrahedron. The volume of a tetrahedron with vertices \( P(x_1, y_1, z_1) \), \( Q(x_2, y_2, z_2) \), \( R(x_3, y_3, z_3) \), and \( S(x_4, y_4, z_4) \) can be found using the determinant formula. \[ V = \frac{1}{6} \left| \begin{vmatrix} x_1 & y_1 & z_1 & 1
x_2 & y_2 & z_2 & 1
x_3 & y_3 & z_3 & 1
x_4 & y_4 & z_4 & 1 \end{vmatrix} \right| \]
Step 2: Conclusion. Thus, the volume of the tetrahedron is \( \frac{2}{3} \).