Question

state paralellogram law of vectors derive an expression for the magnitude and direc
tion of the resultant vector ​

Answer 1

Explanation:

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Answer 2

Answer:

If two vectors are represented both in Magnitude and Direction by the two adjacent sides of a parallelogram drawn from a point, then their resultant is represented both in Magnitude & Direction by the diagonal passing through the same point.

Expression: Consider P and Q from from O as a common initial point.
Let: θ be the angle between the two vectors P and Q. The horizontal component Q is AD = Qcosθ

∴ The vertical component of Q is CS = Q sinθ

⇒ From the right angled triangle,

⇒ ΔODC, OC² = OD² + CD² [∵ (Hypotenuse)² = (Side)² × (Side)²]

= R² = (OA + AD)² + CD²

= R² = (P + Qcosθ)² + (Q sinθ)²

= R² = P² + Q² cos²θ + Q² sin²θ + 2PQcosθ

= R² = P² + Q² + 2PQcosθ

= R = $\sqrt{P^2 + Q^2 +2PQcosθ}$ [I = θ]

⇒ If the Direction of Resultant R makes an angle α (alpha) with the vector P,
= Tanα =  $\frac{CD}{OD}$ = $\frac{CD}{OA+AD}$

= Tanα = Qsinθ/P + Qcosθ

= α = Tan⁻¹ (qsinθ/P + Qcosθ)