Vectors

$\sqrt{9 + 16 + 144} = \sqrt{169} = 13$. Squaring removes the sign of the $-12$.

$(1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8$. The result is a number, not a vector.

The magnitude is 5, so divide each component by 5. Check: $(0.6)^2 + (0.8)^2 = 1$ ✓

The scalar product must be zero: $3 - t + 2 = 0$, so $t = 5$.

Parallel means the same direction vector. The position vector of the given point supplies the starting point.

In three dimensions lines can pass at different depths without touching. A plane offers no such extra direction, so non-parallel lines there always intersect.