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.