Question

Maximum and minimum magnitudes of the resultant of two vectors of magnitudes \[P\] and \[Q\] are in the ratio \[3:1\] . Which of the following relations is true?
(a) \[P = Q\]
(b) \[P = 2Q\]
(c) \[PQ = 1\]
(d) None of these

Answer 1

Hint: We start from the relation given to us. We then find out when the magnitude is maximum and minimum and during what combination. If we add and subtract the vectors from each other, what will the values be.

Complete answer:
We are given two vectors \[P\] and \[Q\] such that the resultant of the two are in the ratio \[3:1\]
The highest value or the maximum value of the interaction between two vectors would be their sum and the lowest value or the minimum value would be their difference.
Let the resultant of the two vectors when added, be three times x and the resultant of the two vectors when subtracted be one time x.
That is, $P + Q = 3x$
$P - Q = x$
Now, we add both the equations together. So, we get,
\[\left( {P + Q} \right) + \left( {P - Q} \right) = 3x + x\]
Simplifying the equation, we get,
\[ \Rightarrow 2P = 4x\]
dividing both sides by two, we get,
\[ \Rightarrow P = 2x\]
Now, putting the value of P in any one of the equations, we get,
\[ \Rightarrow Q = 3x - 2x\]
Simplifying further,
\[ \Rightarrow Q = x\]
Hence, we get,
$P = 2x$ and $Q = x$

Hence, the correct option is (b) \[P = 2Q\]

Note:
The formula for finding the vector sum of two given vectors is $A + B = \sqrt {{{\left| A \right|}^2} + {{\left| B \right|}^2} + 2\left| A \right|\left| B \right|\cos \theta } $ where $\theta$ is the angle between the two vectors . The maximum value of the resultant of the two vectors is attained when the two vectors are in the same direction. The minimum value of the resultant vector is obtained when the two vectors are in the opposite direction.