If A=5a_x +3a_y +2a_z B=-a_x+4a_y +6a_z C=8a_x +2a_y find the values of alpha and beta such that alpha A + beta B +C is parallel to the y-axis.

If $A=5{a}_{x}+3{a}_{y}+2{a}_{z}$
$B=-{a}_{x}+4{a}_{y}+6{a}_{z}$
$C=8{a}_{x}+2{a}_{y}$
find the values of $\alpha$ and $\beta$ such that $\alpha A+\beta B+C$ is parallel to the y-axis.
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nedervdq3
$\alpha A+\beta B+C=<5\alpha -\beta +8,3\alpha +4\beta +2,2\alpha +6\beta >$
for this vector to be parallel to y-axis
we must have:
$5\alpha -\beta +8=0$ (1)
$2\alpha +6\beta =0$ (2)
(1)*6+(2)
$⇒32\alpha +48=0$
$\alpha =-48/32=-1.5$
use (2)
$2\ast \left(-1.5\right)+6\beta =0$
$⇒\beta =3/6=0.5$
so
$\alpha =-1.5,\beta =0.5$
will make $\alpha A+\beta B+C=<5\alpha -\beta +8,3\alpha +4\beta +2,2\alpha +6\beta >$ parallel to y-axis.