Question

The resultant force of \[5\,N\] and \[10\,N\] cannot be:
A. \[12\,N\]
B. \[8\,N\]
C. \[4\,N\]
D. \[5\,N\]

Answer 1

Hint: We are asked to find the value out of the four options given, which will not be in the range of the resultant of the two vectors. Here, we use the concept of “range of vectors”. This gives us all the possible values that the resultant of two vectors will have in the manner of a set. We use the formula of range and find the range of the resultant vector of \[5N\] and \[10N\]

Formulas used:
The formula used to find the range of the resultant of two vectors is given by,
\[\left| {p - q|} \right. \leqslant r \leqslant |p + q|\]
Where \[p\], \[q\] are the two vectors and \[r\] is the resultant of the two vectors.

Complete step by step answer:
We can start by writing the values given in the question.
The value of the first vector is given as, \[p = 5\,N\]
The value of the second vector is given as, \[q = 10\,N\]
Now we substitute this on the formula to find the range of the two vectors,
\[\left| {p - q|} \right. \leqslant r \leqslant |p + q| = |5 - 10| \leqslant r \leqslant |5 + 10| \\
\therefore \left| {p - q|} \right. \leqslant r \leqslant |p + q|= | - 5| \leqslant r \leqslant |15|\]
In conclusion, the range of this vector is in between, \[5\] and \[15\].

Hence, the correct answer is C.

Note: We can also do this by the method of vector addition. The upper limit will be the highest value or when the two vectors are parallel to each other (that is, moves in the same direction) and the lower limit will be the lowest value which will be when the two vectors are antiparallel to each other (that is, in the opposite direction). The formula for the resultant of two vectors is,
\[{F_1}^2 + {F_2}^2 + {F_1}{F_2}\cos \theta \]
Where \[{F_1}\] and \[{F_2}\] are the individual vectors and \[\theta \] is the angle between them (zero degrees if parallel and one hundred and eighty if antiparallel).