Question

drive an expression for the law of parallelogram of vector addition please give me ans
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Answer 1

GiveN:-

• Making proper diagram and writing mathematical expression, state parallelogram law of vector addition.

SolutioN :-

Let us assume two vectors acting at an angle $\theta$ to each other . Their tails are joined end to end . According to the vector law of Parallelogram addition , if two letters are represented by two adjacent sides of a parallelogram then the resultant is given by the diagonal of the parallelogram . Let us take , that ,

$\red{\frak{ Let }}\begin{cases} \textsf{ First vector be \sf \vec{A} .} \\\textsf{ Second vector be \sf \vec{B} .}\end{cases}$

Also the components on $\sf \vec{A}$ will be $\sf \vec{A}sin\theta \ \& \ \vec{A}cos \theta$

★ Using Pythagoras Theorem in ∆AEC .

$\sf:\implies \pink{ Hypotenuse^2 = perpendicular^2+base^2 }\\\\\sf:\implies \vec{AC}^2= CE^2+AE^2\\\\\sf:\implies \vec{AC}= \sqrt{ (A sin\theta)^2+(B + A sin\theta)^2 }\\\\\sf:\implies \vec{AC}= \sqrt{ A^2 sin^2\theta + B^2+A^2 cos^2\theta + 2ABcos\theta} \\\\\sf:\implies \vec{AC} =\sqrt{ A^2(sin^2\theta+cos^2\theta)+B^2+2AB cos\theta } \\\\\sf:\implies\underset{\blue{\sf Required \ Answer }}{\underbrace{ \boxed{\pink{\frak{ \vec{AC}= \sqrt{ \vec{A}^2 +\vec{B}^2+2\vec{A}\vec{B} cos\theta }}}}}}$