Answer:
The value of [tex]k[/tex] is -9.
Step-by-step explanation:
If [tex]A(x,y) = (-1,-6)[/tex], [tex]B(x,y) =(3, -12)[/tex] and [tex]C(x,y) = (k,6)[/tex], then [tex]\overrightarrow{BC} = \alpha\cdot \overrightarrow{BA}[/tex]. By vectors sum, we find each vector below:
[tex]\overrightarrow{BA} = A(x,y)-B(x,y)[/tex] (1)
[tex]\overrightarrow{BA} = (-1,-6)-(3,-12)[/tex]
[tex]\overrightarrow{BA} = (-4, 6)[/tex]
[tex]\overrightarrow{BC} = C(x,y)-B(x,y)[/tex] (2)
[tex]\overrightarrow{BC} = (k,6)-(3,-12)[/tex]
[tex]\overrightarrow{BC} = (k-3, 18)[/tex]
By substituting in the equation defined at the begining of this answer:
[tex](k-3, 18) =\alpha\cdot (-4, 6)[/tex]
[tex](k-3, 18) = (-4\cdot \alpha, 6\cdot \alpha)[/tex] (3)
The value of [tex]\alpha[/tex] is:
[tex]6\cdot \alpha = 18[/tex] (3a)
[tex]\alpha = 3[/tex]
If we know that [tex]\alpha = 3[/tex], then the value of [tex]k[/tex] is:
[tex]k-3 = -4\cdot \alpha[/tex]
[tex]k = 3-4\cdot \alpha[/tex]
[tex]k = 3-4\cdot (3)[/tex]
[tex]k = 3-12[/tex]
[tex]k = -9[/tex]
Then, the value of [tex]k[/tex] is -9.
