If ` vec a, vec b , vec c ` are three non- coplanar vectors such that ` vec a + vec b + vec c = alpha vec d and vec b +vec c + vec d = beta vec a, " then " vec a + vec b + vec c + vec d ` to equal to

Options

A. `vec 0`
B. `alpha vec a `
C. `beta vec b`
D. `(alpha+ beta) vec c `

Answer

To solve the problem step by step, we will use the given equations involving the vectors \( \vec{a}, \vec{b}, \vec{c}, \) and \( \vec{d} \).

### Step 1: Write down the given equations
We are given:
1. \( \vec{a} + \vec{b} + \vec{c} = \alpha \vec{d} \) (Equation 1)
2. \( \vec{b} + \vec{c} + \vec{d} = \beta \vec{a} \) (Equation 2)

### Step 2: Express \( \vec{d} \) in terms of the other vectors
From Equation 2, we can isolate \( \vec{d} \):
\[ \vec{d} = \beta \vec{a} - \vec{b} - \vec{c} \]

### Step 3: Substitute \( \vec{d} \) into Equation 1
Now substitute \( \vec{d} \) from Step 2 into Equation 1:
\[ \vec{a} + \vec{b} + \vec{c} = \alpha (\beta \vec{a} - \vec{b} - \vec{c}) \]

### Step 4: Expand the equation
Expanding the right-hand side, we have:
\[ \vec{a} + \vec{b} + \vec{c} = \alpha \beta \vec{a} - \alpha \vec{b} - \alpha \vec{c} \]

### Step 5: Rearrange the equation
Now, rearranging gives:
\[ \vec{a} + \vec{b} + \vec{c} - \alpha \beta \vec{a} + \alpha \vec{b} + \alpha \vec{c} = 0 \]
This simplifies to:
\[ (1 - \alpha \beta) \vec{a} + (1 + \alpha) \vec{b} + (1 + \alpha) \vec{c} = 0 \]

### Step 6: Use the non-coplanarity condition
Since \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, the only solution to the equation above is:
1. \( 1 - \alpha \beta = 0 \)
2. \( 1 + \alpha = 0 \)

From \( 1 + \alpha = 0 \), we find:
\[ \alpha = -1 \]

Substituting \( \alpha = -1 \) into \( 1 - \alpha \beta = 0 \):
\[ 1 - (-1) \beta = 0 \implies 1 + \beta = 0 \implies \beta = -1 \]

### Step 7: Substitute \( \alpha \) and \( \beta \) back into the equations
Now substitute \( \alpha = -1 \) and \( \beta = -1 \) back into Equation 2:
\[ \vec{b} + \vec{c} + \vec{d} = -\vec{a} \]
This implies:
\[ \vec{d} = -\vec{a} - \vec{b} - \vec{c} \]

### Step 8: Find \( \vec{a} + \vec{b} + \vec{c} + \vec{d} \)
Now, we can find \( \vec{a} + \vec{b} + \vec{c} + \vec{d} \):
\[ \vec{a} + \vec{b} + \vec{c} + \vec{d} = \vec{a} + \vec{b} + \vec{c} + (-\vec{a} - \vec{b} - \vec{c}) = 0 \]

### Final Answer
Thus, the value of \( \vec{a} + \vec{b} + \vec{c} + \vec{d} \) is:
\[ \boxed{0} \]