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?

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?

Philoid · Accepted Answer

A. Incorrect

Explanation: It is not necessary that the vectors ,, and be null vectors for their sum to be a null vector. There can be other combinations as well. For example, if then

.

B. Correct

Explanation:

⇒

Taking modulus on both sides,

⇒

Hence, magnitude of () is equal to the magnitude of ().

C. Correct

Explanation:

⇒

Taking modulus on both sides,

⇒

Hence, the magnitude of is always equal to the magnitude of and can never be greater than that.

D. Correct

Explanation: For the sum to be a null vector, the vectors , and must be three sides of a triangle according to triangle law of vector addition. The three sides of a triangle lie on the same plane. Hence, the vectors , and must be coplanar. But if and are collinear, then must lie on the same line and in opposite direction in order to cancel out in the sum .

NOTE: Triangle law of vector addition states that when two vectors are represented by two sides of a triangle in magnitude and direction taken in same order then third side of the third side of that triangle represents in magnitude and direction the resultant of the vectors.