(a) Incorrect

In order to make 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 that 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 |
$\left|a\right|\le \left|a\right|+\left|b\right|+\left|c\right| .......\left(i\right)$

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

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.