Question

find the magnitude and direction of the resultant of two vectors A and B in terms of their magnitudes and angle theta between them

Answer 1

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Hope that the answer helps you. See the pic above I hope that u'll be able to understand.^_^.

Answer 2

Answer: $\theta = tan ^{-1} b sin \alpha / a + b cos \alpha.$

Solution:

To know the resultant vector magnitude and its angle between, simply apply the triangle law of vector addition. Assume the two vector A and B with an angle alpha whose resultant be vector C given by vector C = vector A + vector B. In addition drop a line from the point B and C meeting to form the right angled triangle, so $AC ^2 = AD ^2 + CD ^2$

=> $AC = \sqrt{ AD ^2 + CD ^2 }$

=> $AC = \sqrt{ ( AB + BD) ^2 + CD ^2}$

=> $AC = \sqrt{ Ab ^2 + 2 AB x BD + BD ^2 + CD ^2 }$

=> From triangle BCD, cos alpha = BD / BC

=> $BD = BC \times cos \alpha$

=> $AC = \sqrt{a^2 + 2 ab cos \alpha + BC ^2}$

=> $AC = \sqrt{a ^2 + b ^2 + 2 a b cos \alpha}$

=> $c = \sqrt{a ^2 + b ^2 + 2 a b cos \alpha}$

The resultant vector makes angle theta therefore, in triangle BCD,

    $sin \alpha = CD / BC$

=> $CD = BC sin \alpha,$

and  in triangle ADC,

$tan \theta = CD / AD = BC sin \alpha / a + b cos \alpha$

=> $\theta = tan ^{-1} b sin \alpha / a + b cos \alpha.$