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 magnitude 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

Hii dear,

# Answers with explainations-
a) Incorrect.
It's not necessary for all four vectors to be null vectors for sum to be null vectors, just having different directions canceling each others effect is enough.

b) Correct.
Given a+b+c+d = 0
a+b = -(c+d)
In terms of magnitude,
|a+b| = |c+d|

c) Correct.
Given is a+b+c+d = 0
a = -(b+c+d)
In terms of magnitude,
|a| = |b+c+d|
|a| <= |b| + |c| + |d|
Hence, magnitude of a will always be lesser than or equal to sum of magnitudes of b, c and d.

d) Correct.
Given is a+b+c+d = 0
This is possible only if all vectors are either coplanar or collinear.

Hope that was useful...

Answer 2

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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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