Find the value of m such that the vector (𝑚, 7, -4) is a linear combination of vectors (-2, 2, 1) 𝑎𝑛𝑑 (2, 1, -2)

To find the value of m such that the vector (m, 7, -4) is a linear combination of vectors (-2, 2, 1) and (2, 1, -2), we need to solve the following system of equations:

-2a + 2b = m 2a + b = 7 a - 2b = -4

where a and b are the scalar multiples of the vectors (-2, 2, 1) and (2, 1, -2) respectively.

First, let's solve the second and third equations for a and b.

From the second equation, we can express a in terms of b:

a = (7 - b) / 2

Substitute a into the third equation:

(7 - b) / 2 - 2b = -4 7 - b - 4b = -8 -5b = -15 b = 3

Substitute b = 3 into the equation for a:

a = (7 - 3) / 2 a = 2

Now, substitute a = 2 and b = 3 into the first equation:

-22 + 23 = m -4 + 6 = m m = 2

So, the value of m such that the vector (m, 7, -4) is a linear combination of vectors (-2, 2, 1) and (2, 1, -2) is 2.