# Find angle BCC¹ on triangular prism

In a triangular prism $$ABCA¹B¹C¹$$, $$|AB|=c$$, $$|BC|=a$$, $$|CA|=b$$, $$\angle BAA¹=\alpha$$, $$\angle CAA¹=\beta$$. Find $$\angle BCC¹$$

Vector methods are preferred.

• Please show your effort before entering the question. – Narasimham May 26 at 18:54

$$BC=AC-AB$$, $$CC^1=AA^1=BB^1$$,
$$\cos\angle BCC^1=\frac{BC.CC^1}{|BC|\cdot|CC^1|} =\frac{(AC-AB).CC^1}{|BC|\cdot|CC^1|} =\frac{AC.CC^1-AB.CC^1}{|BC|\cdot|CC^1|} =\frac{AC.AA^1-AB.AA^1}{|BC|\cdot|CC^1|} =\frac{|AC|\cdot|AA^1|\cdot\cos\angle CAA^1 -|AB|\cdot|AA^1|\cdot\cos\angle BAA^1}{|BC|\cdot|CC^1|} =\frac{|AC|\cdot\cos\angle CAA^1 -|AB|\cdot\cos\angle BAA^1}{|BC|} =\frac{b\cos\beta-c\cos\alpha}{a}$$