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Using	the	Scalar	Product	to	Prove	the	Cosine	Rule		
	
	
Part	A		
Before	we	start	this	proof	you	need	to	use	the	following	properties	of	the	scalar	
product:	
	
1. 𝒄 ∙ 𝒄 = 	
	
2. 𝒃 ∙ 𝒄 = 	
	
3. 𝒃 − 𝒄 ∙ (𝒃 − 𝒄) = 	
	
	
	
Part	B	
Label	the	following	triangle	ABC	where:	
	
𝐴𝐵	represents	the	vector		𝒄	
	
𝐴𝐶	represents	the	vector		𝒃	
	
What	is	𝐵𝐶	in	terms	of	𝒄 and	𝒃?	Label	this	on	the	diagram		
	
What	is	𝐵𝐶 ∙ 𝐵𝐶	?	
	
Using	1,2	and	3	from	part	A	put	this	altogether	to	find	the	length	of	(𝐵𝐶)!

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Using the scalar product to prove the cosine rule