Question

Which of the sets given below may represent the magnitudes of the three vectors adding to zero?
A) $2,4,8$
B) $4,8,16$
C) $1,2,1$
D) $0.5,1,2$

Answer 1

It is known to us that a vector is a physical quantity that has both magnitude and direction. But to find the magnitude of the vector, its length needs to be calculated. As the magnitudes are already given in the options, therefore we need to find the resultant vector of any two vectors and compare it with the third vector. To get the answer, we must keep in mind the direction of the vectors.

Complete answer:
To find the resultant vector, we just have to add any two vectors from the given sets. We take the first set, $2,4,8$ . Let the magnitude of the first vector is $2$ and the second vector be $4$.The resultant vector’s magnitude is,
$2 + 4 = 6$
For the sum to be zero, the magnitude of the resultant vector of any two vectors should be equal to the magnitude of the third vector. But here the magnitudes are not equal. Therefore this option is incorrect.
Similarly, for sets 2 and 4, the sum of the first two vectors gives:
$4 + 8 = 12$
$0.5 + 1 = 1.5$
Here also we observe that the sum of two vectors is not equal to the third vector and hence cannot be added up to zero.
Now, take set 3 which is $(1,2,1)$ . Let the magnitude of the first vector and second vector be $1$ each. So, the sum of the vectors would be:
$1 + 1 = 2$
Now as the sum of the vectors comes out to be equal to the magnitude of the third vector, we must assume their directions now. Let the first and the second vector be in the same direction and the third vector is in the opposite direction. Therefore the magnitude of the overall results would be:
$- 2 + 2 = 0$
Hence, the sum comes out to be zero for set $3$ , and therefore option (C) is correct.

Note:
To find the resultant vector of the two vectors, we can use triangle law of addition which states that when two vectors are represented as two sides of a triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector. It is known to us that the sum of any two sides of a triangle is equal to or greater than the third side of the triangle. So, by using these two things, we can also find the answer to the question.