I have two exercises here that are giving me problems. I don't have a strong enough conceptual grasp on the material I guess.

Find the value k so that the points A=(2,-1), B=(1,4), and C=(k,9) are aligned.

Find the values of a and b so that the two lines r: ax-y+2=0 and s: bx+6y-9=0 are perpendicular and also so that the second line passes through the point P=(1,1).

Additionally, if anyone has a recommendation for sources to brush up on these kinds of exercises, I feel they would greatly strengthen my comprehension for the linear algebra class I'm taking.

That said, if there are multiple paths to solutions to these exercises, ideally I would like to relate them to concepts such as vectors, dot product, etc. Thank you again.


Hint: For one, calculating the line passing through $A$ and $B$, this is $$y=-5x+9$$ then for $C$ is hold: $$9=-5k+9$$ For second, we get plugging the coordinates in the given equation $$b+6-9=0$$ so $b=3$ and the equation is $$y=-\frac{1}{2}x+\frac{3}{2}$$ So the slope of the perpendicular line is given by $$2$$


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