Question

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?

Answer 1

(a) Incorrect

In order to make a + b + c + d = 0, it is not necessary to have all four given vectors to be null vectors. There are many other combinations which can give the sum zero.

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(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). 

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(c) Correct 

a + b + c + d = 0 a = (b + c + d) ...(i)Taking modulus both sides, we get: 
| a | = | b + c + d |  
|a| ≤ |a| + |b| + |c|……………. (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.

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(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.