__Question 4.7__:

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 ?

__Solution__:

**(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)**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 |**≤

**| 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

**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.

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