Find angle BCC¹ on triangular prism

Question

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

Vector methods are preferred.

Answer 1

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