Given vectors a (2; 0; -1) and b (3; -1; 2). At what value of K will vectors 2a-kb and b-a be perpendicular?

Knowing the coordinates of vectors a and b, we find the coordinates of vectors 2а – kb and b – а:

2а – kb = (2 * 2 – k * 3; 2 * 0 – k * (-1); 2 * (-1) – k * 2) = (4 – 3k; k; -2 – 2k);

b – a = (3 – 2; -1 – 0; 2 – (-1)) = (1; -1; 3).

Find the scalar product of vectors 2а – kb and b – а:

(2а – kb, b – а) = (4 – 3k) * 1 + k * (-1) + (-2 – 2k) * 3 = 4 – 3k – k – 6 – 6k = -2 – 10k.

Since two vectors will be perpendicular if and only if their dot product is 0, we can make the following equation:

-2 – 10k = 0,

solving which, we get:

10k = -2;

k = -2/10 = -0.2.

Answer: for k = -0.2.