Question

derive expression for the resultant of two concurrent vector

Answer 1

Let two concurrent vectors be P and Q. Let the resultant be R.

R=P+Q

Refer to the diagram attached below,

In triangle OCB,

OB²=OC²+BC²

⇒OB²=(OA+AC)²+BC²

cos θ=$\frac{AC}{AB}$

AC=AB cos θ

AC=OD cos θ=Q

cos θ=[tex]\frac{BC}{AB}[/tex[

BC=AB sin θ

BC=OD sinθ=Q sin θ

Substitute the values in the resultant:

R²=(P+Q cosθ)²+(Q sin θ)²

⇒R²=P²+Q²cos²θ+2PQ cos θ+ Q² sin²θ

⇒R²=P²+Q²(cos²θ+ sin²θ)+2PQ cos θ= P²+Q²+2PQ cos θ

⇒R=√(P²+Q²+2PQ cos θ)

Answer 2

Let two concurrent vectors be P and Q. Let the resultant be R.

R=P+Q

In triangle OCB,

OB²=OC²+BC²

⇒OB²=(OA+AC)²+BC²

cos θ=\frac{AC}{AB}ABAC

AC=AB cos θ

AC=OD cos θ=Q

cos θ=[tex]\frac{BC}{AB}[/tex[

BC=AB sin θ

BC=OD sinθ=Q sin θ

Substitute the values in the resultant:

R²=(P+Q cosθ)²+(Q sin θ)²

⇒R²=P²+Q²cos²θ+2PQ cos θ+ Q² sin²θ

⇒R²=P²+Q²(cos²θ+ sin²θ)+2PQ cos θ= P²+Q²+2PQ cos θ

⇒R=√(P²+Q²+2PQ cos θ)

Hope this helps u