a) Two vectors which have magnitude 8 and 10 can have maximum value of magntude of their resultant as
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Answer:
The magnitude of the resultant of the two vectors a→ and b→ be is given by
∣∣∣a→+b→∣∣∣=∣∣a→∣∣2+∣∣∣b→∣∣∣2+2|a→||b→|cos(a→,b→)−−−−−−−−−−−−−−−−−−−−−−−−−−−√
The maximum and minimum values of this expressions depends on the angle between the two vectors, since the maximum and minimum of the cosine function are +1 and −1 respectively at the angles 0 and π .
The resultant of two vectors is maximum when they are parallel.
∣∣∣a→+b→∣∣∣
=∣∣a→∣∣2+∣∣∣b→∣∣∣2+2|a→||b→|cos(0)−−−−−−−−−−−−−−−−−−−−−−−√
=∣∣a→∣∣2+∣∣∣b→∣∣∣2+2|a→||b→|−−−−−−−−−−−−−−−−−−√
=(|a→∣∣+∣∣∣b→∣∣∣)2−−−−−−−−−−−√
=∣∣a→∣∣+∣∣∣b→∣∣∣
The resultant of two vectors is minimum when they are anti-parallel.
∣∣∣a→+b→∣∣∣
=∣∣a→∣∣2+∣∣∣b→∣∣∣2+2|a→||b→|cos(π)−−−−−−−−−−−−−−−−−−−−−−−√
=∣∣a→∣∣2+∣∣∣b→∣∣∣2−2|a→||b→|−−−−−−−−−−−−−−−−−−√
=(|a→∣∣−∣∣∣b→∣∣∣)2−−−−−−−−−−−√
=∣∣∣∣∣a→∣∣−∣∣∣b→∣∣∣∣∣∣
Given that the magnitudes of the two vectors are 8 and 10.
So, the maximum resultant =∣∣a→∣∣+∣∣∣b→∣∣∣=8+10=18
The minimum resultant =∣∣∣∣∣a→∣∣−∣∣∣b→∣∣∣∣∣∣=|8−10|=2
Explanation:
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