At what value of a the vectors p (4; a) and c (-5; 2) are perpendicular.

The perpendicularity theorem for two nonzero vectors states that for two nonzero vectors to be perpendicular, it is necessary and sufficient that their scalar product be zero, that is, for the equality (vector p, vector c) = 0 to hold.
We calculate the scalar product of vectors p and c in coordinates:
(vector p, vector c) = px * cx + py * cy = 4 * (-5) + a * 2 = -20 + 2a.
Let us find the value of a, at which the perpendicularity condition is satisfied:
-20 + 2a = 0;
2a = 20;
a = 20/2;
a = 10.
Answer: a = 10.