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 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

(a)

The given statement is not correct. Since the addition of the four vectors a → , b → , c → and d → gives the value 0 even if a → =− c → and b → =− d → .

(b)

Given, a+b+c+d=0.

Rearrange the given equation.

a+c=-(b+d) (I)

Take modulus on both sides of the equation (I).

|a+c|=|-(b+d)|

(a+c)=(b+d)…………………(II)

It is clear from equation (II) that the magnitude of (a+c) equals the magnitude of (b+d).

Hence, the given statement is true.

(c)

Given, a+b+c+d=0

Rearrange the given equation.

a =-(b+c+d) (III)

Take modulus on both sides of equation (III).

|a |= |-(b+c+d)|

|a | ≤ |b|+|c|+|d| (IV)

It is clear from equation (IV) that |-(b+c+d)| is the sum of vectors b, c and d. So, the magnitude of |-(b+c+d)| will be less than or equal to the sum of magnitude of vectors b, c and d. Hence, the magnitude of a can never be greater than or equal to sum of the magnitude of vectors b, c and d.

Hence, the given statement is correct.

(d)

The given statement is correct.

If the sum of vector b → and vector c → , that is b → + c → do not lie in the plane of a → + d → , then the vector sum will not be equal to zero because the addends will have different magnitude and direction.