Write the equation of a plan (in any form) through the point (2,-4,1) and parallel to the line [x,y,z]= [3,-2,5]+ t[5,1,-3]

Question

Write the equation of a plan (in any form) through the point (2,-4,1) and parallel to the line [x,y,z]= [3,-2,5]+ t[5,1,-3]

Answer 1

[2, -4, 1] - [3, -2, 5] = [-1, -2, -4]

i j k

-1 -2 -4

5 1 -3

has determinant 10i - 23j + 9k

Then, consider the plane [2, -4, 1] + s[10, -23, 9] + t[5, 1, -3]

Alternatively, we can write the plane via

i j k

10 -23 9

5 1 -3

The determinant is 60i + 75j + 125k, which we may write as 5(12i + 15j + 25k)

Thus, an alternative equation for the plane is

12x + 15y + 25z = 12*2 + 15*-4 + 25 * 1 = -11

12x + 15y + 25z = -11