 # Determine beta and alpha by using vectors such that A, B and C lie in the same plane, given that vector vec(AB)=−4vec(i)−vec(j) −2vec(k) and vector vec(BC)=4 vec(i) +(beta+3)vec(j) +(alpha−6)vec(k) Nica2t 2022-08-13 Answered
Determine $\beta$ and $\alpha$ by using vectors such that A, B and C lie in the same plane, given that vector $\stackrel{\to }{AB}=-4\stackrel{\to }{ı}-\stackrel{\to }{ȷ}-2\stackrel{\to }{k}$ and vector $\stackrel{\to }{BC}=4\stackrel{\to }{ı}+\left(\beta +3\right)\stackrel{\to }{ȷ}+\left(\alpha -6\right)\stackrel{\to }{k}$
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Any three points in 3D space will inevitably be on the same plane.
To show this, you can take:
$\stackrel{\to }{a}=\stackrel{\to }{AB}$
$\stackrel{\to }{b}=\stackrel{\to }{BC}$
$\stackrel{\to }{c}=\stackrel{\to }{AC}=\stackrel{\to }{AB}+\stackrel{\to }{BC}$
and you will get that $\stackrel{\to }{a}\cdot \left(\stackrel{\to }{b}×\stackrel{\to }{c}\right)=0$, regardless of $\alpha$ and $\beta$