Question

If \( \hat{i} - \hat{j} - \hat{k} \), \( \hat{i} + \hat{j} + \hat{k} \), \( \hat{i} + \hat{j} + 2\hat{k} \), and \( 2\hat{i} + \hat{j} \) are the vertices of a tetrahedron, then its volume is:

• A. \( 0 \)
• B. \( \frac{1}{6} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{1}{3} \)

Hint:

To find the volume of a tetrahedron given four points, construct three vectors from one vertex and compute the determinant of their matrix. The volume formula is: \[ V = \frac{1}{6} \left| \mathbf{AB} \cdot (\mathbf{AC} \times \mathbf{AD}) \right|. \]

Answer 1

The Correct Option is D

Step 1: Formula for the Volume of a Tetrahedron
The volume \( V \) of a tetrahedron formed by four vertices given as position vectors \( \mathbf{A} \), \( \mathbf{B} \), \( \mathbf{C} \), and \( \mathbf{D} \) is given by: \[ V = \frac{1}{6} \left| \begin{vmatrix} x_1 & y_1 & z_1
x_2 & y_2 & z_2
x_3 & y_3 & z_3 \end{vmatrix} \right| \] where the three vectors forming the parallelepiped are: \[ \mathbf{AB} = \mathbf{B} - \mathbf{A}, \quad \mathbf{AC} = \mathbf{C} - \mathbf{A}, \quad \mathbf{AD} = \mathbf{D} - \mathbf{A}. \] Step 2: Find the Vectors Given vertices: \[ \mathbf{A} = (\ 1, -1, -1), \quad \mathbf{B} = (\ 1, 1, 1), \quad \mathbf{C} = (\ 1, 1, 2), \quad \mathbf{D} = (\ 2, 1, 0). \] Find the vectors: \[ \mathbf{AB} = (1,1,1) - (1,-1,-1) = (0,2,2). \] \[ \mathbf{AC} = (1,1,2) - (1,-1,-1) = (0,2,3). \] \[ \mathbf{AD} = (2,1,0) - (1,-1,-1) = (1,2,1). \] Step 3: Compute the Determinant \[ \begin{vmatrix} 0 & 2 & 2
0 & 2 & 3
1 & 2 & 1 \end{vmatrix} \] Expanding along the first column: \[ = 0 \times \begin{vmatrix} 2 & 3
2 & 1 \end{vmatrix} - 2 \times \begin{vmatrix} 0 & 3
1 & 1 \end{vmatrix} + 2 \times \begin{vmatrix} 0 & 2
1 & 2 \end{vmatrix}. \] Calculate each determinant: \[ \begin{vmatrix} 2 & 3
2 & 1 \end{vmatrix} = (2 \times 1) - (3 \times 2) = 2 - 6 = -4. \] \[ \begin{vmatrix} 0 & 3
1 & 1 \end{vmatrix} = (0 \times 1) - (3 \times 1) = -3. \] \[ \begin{vmatrix} 0 & 2
1 & 2 \end{vmatrix} = (0 \times 2) - (2 \times 1) = -2. \] Step 4: Compute the Determinant Value \[ = 0 + 2(3) + 2(-2) = 0 + 6 - 4 = 2. \] Step 5: Compute the Volume \[ V = \frac{1}{6} \times |2| = \frac{2}{6} = \frac{1}{3}. \] Thus, the correct answer is \( \mathbf{\frac{1}{3}} \).