ABCD is a regular tetrahedron, A is the origin and B lies on x-axis. ABC lies in the xy-plane `|vec(AB)|=2` Under these conditions, the number of possible tetrahedrons is :

Answer

To solve the problem, we need to analyze the conditions given for the regular tetrahedron ABCD. Let's break it down step by step.

### Step 1: Understand the Position of Points
- Point A is the origin, so \( A(0, 0, 0) \).
- Point B lies on the x-axis, which means it can be represented as \( B(b, 0, 0) \).
- The length of vector \( \vec{AB} \) is given as \( |\vec{AB}| = 2 \). Therefore, \( |b| = 2 \), which gives us two possible coordinates for B: \( B(2, 0, 0) \) or \( B(-2, 0, 0) \).

### Step 2: Determine the Position of Point C
- The points A, B, and C lie in the xy-plane. Hence, point C can be represented as \( C(0, c, 0) \), where \( c \) can take positive or negative values.
- Since ABC is a triangle in the xy-plane, point C can be either above or below the x-axis, leading to two possible values for C: \( C(0, c, 0) \) where \( c \) can be \( +y \) or \( -y \).

### Step 3: Determine the Position of Point D
- Point D is the apex of the tetrahedron and must be above or below the plane formed by triangle ABC. Therefore, point D can be represented as \( D(0, 0, d) \), where \( d \) can also take positive or negative values.
- This gives us two possible values for D: \( D(0, 0, +z) \) or \( D(0, 0, -z) \).

### Step 4: Count the Combinations
1. **For \( B(2, 0, 0) \)**:
- C can be \( (0, +c, 0) \) or \( (0, -c, 0) \) → 2 options.
- D can be \( (0, 0, +d) \) or \( (0, 0, -d) \) → 2 options.
- Total combinations for B at \( (2, 0, 0) \) = \( 2 \times 2 = 4 \).

2. **For \( B(-2, 0, 0) \)**:
- C can again be \( (0, +c, 0) \) or \( (0, -c, 0) \) → 2 options.
- D can be \( (0, 0, +d) \) or \( (0, 0, -d) \) → 2 options.
- Total combinations for B at \( (-2, 0, 0) \) = \( 2 \times 2 = 4 \).

### Step 5: Calculate Total Tetrahedrons
- Total number of tetrahedrons = Combinations for \( B(2, 0, 0) \) + Combinations for \( B(-2, 0, 0) \) = \( 4 + 4 = 8 \).

### Final Answer
The total number of possible tetrahedrons is **8**.

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