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 with example,

(b) The magnitude of (a + c) equals the magnitude of (b + d) with example,

(c) The magnitude of a can never be greater than the sum of the magnitudes of b , c, and d with example,

( 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 with example?

Answer 1

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(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 which can give the sum zero.

(b) Correc

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| ≤ |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.

(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.

I hope, this will help you

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