Grade 12 calculus and vectors question: “Determine the vector equation of a line through (4,5,5) that is perpendicular to the line (x, y,z) = (11,-8,4) + t(3,-1,1)”

The line

runs in the direction of its tangent vector; we can get it by taking its derivative:

Any line that runs perpendicular to this line will have a tangent vector that is orthogonal to above. So construct some vector that satisfies this.

Suppose ; then , and we can pick any two values that satisfy this condition. For instance,

And of course,

(1, 3, 0) • (3, -1, 1) = 3 - 3 + 0 = 0

so and are indeed orthogonal.

Now, the line running in the direction of and passing through the origin can be obtained by scaling

More generally, if you have a direction/tangent vector and some point , the line through is given by