Question

Given, $a + b + c + d =0$, which of the following statements are incorrect?

• 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

Answer: (A)

(a) $a + b + c + d$ can be zero in many ways other than $a , b , c$ and $d$ must each be a null vector, e.g. if the vectors are in different directions, then their resultant will be zero.
(b) Since $a + b + c + d =0$
$ \therefore a + c =-( b + d ) $
or $ | a + c |$
(c) Since $a + b + c + d =0$
$\therefore a =-( b + c + d )$
or $ | a |=| b + c + d |$
Therefore, the magnitude of $a$ is equal to the magnitude of $( b + c + d )$.
As magnitude of $( b + c + d )$ can be equal or less than the sum of magnitudes of $b , c$ and $d$ but can never be greater.
Therefore, $a$ also can never be greater than the sum of the magnitudes of $b , c$ and $d$.
(d) Since $a + b + c + d =0$
or $a +( b + c )+ d =0$
The resultant of three vectors $a ,( b + c )$ and $d$ can be zero only when they lie in a plane and can be represented by the three sides of triangle taken in one order.
If $a$ and $d$ are collinear, then $( b + c )$ must be in the line of $a$ and $d$, only then the vector sum of all the vectors will be zero.
Thus, the statement given in option is incorrect, rest are correct.