Question

Assertion: The minimum number of non-coplanar vectors whose sum can be zero is four.

Reason: The resultant of two vectors of unequal magnitude can be zero.

• A. If both Assertion and Reason are true and the Reason is the correct explanation of the Assertion.
• B. If both Assertion and Reason are true but Reason is not the correct explanation of the Assertion.
• C. If Assertion is true but Reason is false.
• D. If the Assertion and Reason both are false.
• E. If Assertion is false but Reason is true.

Answer 1

The correct option is C

If Assertion is true but Reason is false.

Explanation for the Assertion statement:

1. Let us add a third vector to make all the three vectors co-planar. The third vector can be resolved so that one component is in the plane of vectors formed by the first and second vectors.
2. The other component is perpendicular to this plane. Obviously, the results can not be zero, because two vectors and the resolved component of the third vector are in a plane and the other resolved component is perpendicular to that plane.
3. Now we add a fourth vector which is not in the plane formed by the first two vectors.
4. Now let us resolve the fourth vector so that one component is on the plane formed by the first two vectors and another component is perpendicular to that plane. Now we have two initial vectors, resolved components of the third and fourth vectors are in a plane. By adjusting their magnitudes we can make the resultant vector in the plane, zero.
5. Similarly, we have resolved components of the third and fourth vectors perpendicular to the above-mentioned plane. By adjusting their magnitude it is possible to make the resultant zero.
6. Therefore, the assertion is true.

Conclusion:- minimum number of non-coplanar vectors, whose sum is zero, is four.

The explanation for the Reason statement:

1. The resultant of two vectors of unequal magnitudes cannot be zero for any value of $\theta$.
2. Resultant, $R=\sqrt{A^2+B^2+2\times A\times B\times \cos \theta }$ and this can't be zero for any value of $\theta$.
3. Therefore, the Reason is false.

Thus, the assertion is true and the reason is false.

Hence, option (C) is correct.