Question

Find the equation of the line through (2, 3, 0) perpendicular to the vectors a = (1, 2, 3) and b = (3, 4, 5).

Answer 1

The cross product a×b is a vector perpendicular to both. That cross product is
  (1, 2, 3)×(3, 4, 5) = (-2, 4, -2)

This can have common factors removed and be sign-adjusted to (1, -2, 1), so the equation of the line can be written in parametric form as ...
  (x, y, z) = (2, 3, 0) + t·(1, -2, 1)

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When working with vectors and matrices, a suitable calculator can be very helpful. There are several phone and tablet apps and web sites that can do these calculations for you.