Given vectors a + b + c + d = 0, which of the following statements are correct:
(a) Vectors a, b, c, and d must each be a null vector,
(b) The magnitude of vectors (a + c) equals the magnitude of vectors(b+ d),
(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d,
(d) Vectors 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?
Answers
Answer:
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.
(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 | .... (i)
Equation (i) 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.
Answer:
Refers to the attachment
