If the verticles of a tetrahedron have the position vectors `vec0, hati+hatj, 2hatj-hatk` and `hati+hatk` then the volume of the tetrahedron is

Options

A. `(1)/(6)`
B. 1
C. 2
D. 3

Answer

To find the volume of the tetrahedron with given vertices, we can use the formula for the volume of a tetrahedron formed by vectors originating from a common vertex. The formula is:

\[ \text{Volume} = \frac{1}{6} | \vec{AB} \cdot (\vec{AC} \times \vec{AD}) | \]

Where:
- \(\vec{A}\), \(\vec{B}\), \(\vec{C}\), and \(\vec{D}\) are the position vectors of the vertices of the tetrahedron.

### Step 1: Identify the position vectors
Given the vertices of the tetrahedron:
- \(\vec{A} = \vec{0} = (0, 0, 0)\)
- \(\vec{B} = \hat{i} + \hat{j} = (1, 1, 0)\)
- \(\vec{C} = 2\hat{j} - \hat{k} = (0, 2, -1)\)
- \(\vec{D} = \hat{i} + \hat{k} = (1, 0, 1)\)

### Step 2: Calculate the vectors \(\vec{AB}\), \(\vec{AC}\), and \(\vec{AD}\)
- \(\vec{AB} = \vec{B} - \vec{A} = (1, 1, 0) - (0, 0, 0) = (1, 1, 0)\)
- \(\vec{AC} = \vec{C} - \vec{A} = (0, 2, -1) - (0, 0, 0) = (0, 2, -1)\)
- \(\vec{AD} = \vec{D} - \vec{A} = (1, 0, 1) - (0, 0, 0) = (1, 0, 1)\)

### Step 3: Calculate the cross product \(\vec{AC} \times \vec{AD}\)
To find the cross product \(\vec{AC} \times \vec{AD}\):

\[ \vec{AC} = (0, 2, -1), \quad \vec{AD} = (1, 0, 1) \]

Using the determinant method for the cross product:

\[ \vec{AC} \times \vec{AD} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 2 & -1 \\ 1 & 0 & 1 \end{vmatrix} \]

Calculating this determinant:

\[ = \hat{i} \begin{vmatrix} 2 & -1 \\ 0 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 0 & -1 \\ 1 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 0 & 2 \\ 1 & 0 \end{vmatrix} \]

Calculating each of these 2x2 determinants:

1. \(\begin{vmatrix} 2 & -1 \\ 0 & 1 \end{vmatrix} = (2)(1) - (0)(-1) = 2\)

2. \(\begin{vmatrix} 0 & -1 \\ 1 & 1 \end{vmatrix} = (0)(1) - (-1)(1) = 1\)

3. \(\begin{vmatrix} 0 & 2 \\ 1 & 0 \end{vmatrix} = (0)(0) - (2)(1) = -2\)

Putting it all together:

\[ \vec{AC} \times \vec{AD} = 2\hat{i} - 1\hat{j} - 2\hat{k} = (2, -1, -2) \]

### Step 4: Calculate the dot product \(\vec{AB} \cdot (\vec{AC} \times \vec{AD})\)
Now we calculate the dot product:

\[ \vec{AB} \cdot (\vec{AC} \times \vec{AD}) = (1, 1, 0) \cdot (2, -1, -2) \]

Calculating this:

\[ = (1)(2) + (1)(-1) + (0)(-2) = 2 - 1 + 0 = 1 \]

### Step 5: Calculate the volume of the tetrahedron
Now we can find the volume:

\[ \text{Volume} = \frac{1}{6} |1| = \frac{1}{6} \]

### Final Answer
The volume of the tetrahedron is \(\frac{1}{6}\).

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