Question

Given $\overline{a}+\overline{b}+\overrightarrow{c}+\overline{d}=0$ , which of the following statements is/are not a correct statement?

• A. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ must be a null vector.
• B. The magnitude of $(\overrightarrow{a}+\overrightarrow{c})$ equals the magnitude of $a(\overrightarrow{b}+\overrightarrow{d})$
• C. The magnitude of $\overrightarrow{a}$ can never be greater than the sum of the magnitudes of $\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$
• D. $\overrightarrow{b}$+$\overrightarrow{c}$ must He in the plane of $\overrightarrow{a}$ and $\overrightarrow{d}$ if $\overrightarrow{a}$ and $\overrightarrow{d}$ are not collinear and in the line of $\overrightarrow{a}$ and $\overrightarrow{d}$, if they are collinear.

Answer 1

The correct option is A $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ must be a null vector.

In order to make vectors 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.

Simple example:- a=î,b=2î,c=–3î,d=0

(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 | …. (#)

Equation (#) 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.

This is the explanation of correct solution.