Question

Given $\text{a + b + c + d = 0}$, which of the following statements are correct.
(a) $\text{a, b, c, d}$ must each be a null vector,
(b) The magnitude of ($\text{a+c}$) equals the magnitude of ($\text{b + d}$),
(c) The magnitude of $\text{a}$ can never be greater than the sum of the magnitudes of $\text{b, c}$ and $\text{d}$,
(d) $\text{b + c}$ must lie in the plane of $\text{a}$ and $\text{d}$ if $\text{a}$ and $\text{d}$ are not collinear, and in the line of $\text{a}$ and $\text{d}$, if they are collinear ?

Answer 1

(a) In order to make vectors a + b + c + d = 0, it is not necessary to have all the four given vectors to be null vectors. There are many other combinations which can give the sum zero.
(b) Correct
a + b + c + d = 0
a + c = -(b + d)
Taking modulus on both the sides, we get:
| a + c | = | -(b + d)| = | b + d |
Hence, the magnitude of (a + c) is the same as the magnitude of (b + d).
(c) Correct
a + b + c + d = 0
a = - (b + c + d)
Taking modulus both sides, we get:
| a | = | b + c + d |
| a | $\le$ | b | + | c | + |d| .... (i)

Equation (i) shows that the magnitude of a is equal to or less than the sum of the magnitudes of b, c, and d.
Hence, the magnitude of vector a can never be greater than the sum of the magnitudes of b, c, and d.

(d) Correct
For a + b + c + d = 0
a + (b + c) + d = 0
The resultant sum of the three vectors a, (b + c), and d can be zero only if (b + c) lie in a plane containing a and d, assuming that these three vectors are represented by the three sides of a triangle.
If a and d are collinear, then it implies that the vector (b + c) is in the line of a and d. This implication holds, only then the vector sum of all the vectors will be zero.