If the points with position vectors  αiˆ+10jˆ+13kˆ, 6iˆ+11jˆ+11kˆ,92iˆ+βjˆ−8kˆ are collinear, then (19α−6β)2 is equal to

The given points are collinear, which means they lie on the same line. We can express this condition using the concept of vectors. If A, B, and C are three collinear points, then the vectors AB and BC are parallel.

The position vectors of the points are given as:

A = αi + 10j + 13k B = 6i + 11j + 11k C = 92i + βj - 8k

We can find the vectors AB and BC as follows:

AB = B - A = (6 - α)i + (11 - 10)j + (11 - 13)k = (6 - α)i + j - 2k BC = C - B = (92 - 6)i + (β - 11)j + (-8 - 11)k = 86i + (β - 11)j - 19k

For AB and BC to be parallel, the ratios of their corresponding components must be equal. Therefore, we have:

(6 - α) / 86 = 1 / (β - 11) = -2 / -19

Solving these equations, we get:

α = 6 - 86/19 = 6 - 4.526 = 1.474 β = 11 + 19 = 30

Substituting these values into the expression (19α - 6β)², we get:

(19 * 1.474 - 6 * 30)² = (-45.056)² = 2030.053136

So, (19α - 6β)² is approximately equal to 2030.053136.