Given a + b + c + d = 0, which of the following statements are correct:
A. a, b, c, and d must 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 collinear, and in the line of a and d, if they are collinear?
A. Incorrect
Explanation: It is not necessary that the vectors 
,
,
and 
be null vectors for their sum to be a null vector. There can be other combinations as well. For example, if 
then
.
B. Correct
Explanation:
⇒ 
Taking modulus on both sides,
⇒ 
Hence, magnitude of (
) is equal to the magnitude of (
).
C. Correct
Explanation:
⇒ 
Taking modulus on both sides,
⇒ 
Hence, the magnitude of
is always equal to the magnitude of 
and can never be greater than that.
D. Correct
Explanation: For the sum 
to be a null vector, the vectors 
,
and 
must be three sides of a triangle according to triangle law of vector addition. The three sides of a triangle lie on the same plane. Hence, the vectors 
,
and 
must be coplanar. But if 
and 
are collinear, then
must lie on the same line and in opposite direction in order to cancel out 
in the sum 
.
NOTE: Triangle law of vector addition states that when two vectors are represented by two sides of a triangle in magnitude and direction taken in same order then third side of the third side of that triangle represents in magnitude and direction the resultant of the vectors.