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

Vector methods are preferred.

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  • $\begingroup$ Please show your effort before entering the question. $\endgroup$ – Narasimham May 26 at 18:54
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$BC=AC-AB$, $CC^1=AA^1=BB^1$,
$\cos\angle BCC^1=\frac{BC.CC^1}{|BC|\cdot|CC^1|} =\frac{(AC-AB).CC^1}{|BC|\cdot|CC^1|} =\frac{AC.CC^1-AB.CC^1}{|BC|\cdot|CC^1|} =\frac{AC.AA^1-AB.AA^1}{|BC|\cdot|CC^1|} =\frac{|AC|\cdot|AA^1|\cdot\cos\angle CAA^1 -|AB|\cdot|AA^1|\cdot\cos\angle BAA^1}{|BC|\cdot|CC^1|} =\frac{|AC|\cdot\cos\angle CAA^1 -|AB|\cdot\cos\angle BAA^1}{|BC|} =\frac{b\cos\beta-c\cos\alpha}{a}$

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