Given `a+b+c+d=0`. Which of the following statements(s) is (are) correct?

Options

A. `a,b,c,` and `d` much each be a null vector
B. The magnitude of `a+c` equals the magnitude of `b+d`
C. The magnitude of `a` can never be greater than the sum of the magnitudes of `b,c` and `d`
D. `b+c` must lie in the plane of `a` and `d` if `a` and `d` are not colinear and in the line of `a` and `d` if they are collinear

Answer

To solve the problem given the equation \( a + b + c + d = 0 \), we need to analyze the statements provided and determine which are correct.

### Step 1: Understand the equation
The equation \( a + b + c + d = 0 \) implies that the vector sum of vectors \( a \), \( b \), \( c \), and \( d \) equals the null vector (zero vector). This means that the vectors can be arranged in such a way that they cancel each other out.

### Step 2: Analyze each option

**Option A:** "A, B, C, and D must each be equal to the null vector."
- This statement is **incorrect**. While it's true that the sum of the vectors equals the null vector, it does not imply that each vector must be the null vector. For instance, \( a \), \( b \), \( c \), and \( d \) could be non-zero vectors that still sum to zero.

**Option B:** "The magnitude of A + C equals the magnitude of B + D."
- This statement is **correct**. From the equation \( a + b + c + d = 0 \), we can rearrange it to \( a + c = - (b + d) \). The magnitudes of both sides are equal, meaning \( |a + c| = |b + d| \). The negative sign indicates that they are in opposite directions, but their magnitudes are the same.

**Option C:** "The magnitude of vector A can never be greater than the sum of the magnitudes of A, B, C, and D."
- This statement is **correct**. From the equation \( a = - (b + c + d) \), we can conclude that the magnitude of \( a \) cannot exceed the sum of the magnitudes of \( b \), \( c \), and \( d \) because they are in opposite directions. Thus, \( |a| \leq |b| + |c| + |d| \).

**Option D:** "B + C must lie in the plane of A and D if A and D are non-collinear."
- This statement is **correct**. If vectors \( a \) and \( d \) are non-collinear, then the sum \( b + c \) must lie in the plane formed by \( a \) and \( d \). This is due to the vector addition and the properties of vector spaces.

### Conclusion
Based on the analysis:
- **Option A** is incorrect.
- **Options B, C, and D** are correct.

### Final Answer
The correct statements are B, C, and D.

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