vectors and scalars - 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 ?

Grigori (last edited 5 months ago) (Student, UG2 Grade)

(a) Incorrect statement

There are several other combinations that can result in the sum being zero(a + b + c + d = 0). It is not mandatory for all four vectors to be null vectors for the sum to be null vectors.

(b) Correct statement

a + b + c + d = 0

a + b = - (c + d)

Taking modulus on both sides yields magnitude

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

|a + b| = |c + d|

Hence, the magnitude of (a + c) equals the magnitude of (b + d).

a + b + c + d = 0

a = - (b + c + d)

Taking modulus on both sides yields magnitude

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

|a| =|b + c + d|

The sum of the magnitudes of b, c, and d is equal to or less than the magnitude of a.

Hence, the magnitude of vector a will never exceed the sum of the magnitudes of vectors b, c, and d.

d) Correct statement

a + b + c + d = 0

a + (b + c) + d = 0

The sum of the three vectors a, (b+c) and d can only be nullified if (b+c) lies on a plane containing a and d,

If a and d are collinear, the vector (b + c) is in line with a and d, then the vector sum of all the vectors become zero.