Quick questions on Vector equations of lines and planes: VCE Specialist Mathematics Unit 3

Substitute the parametric coordinates of the line into the plane's Cartesian equation. This gives one equation in $t$; solve it, then back-substitute to get the intersection point. If the coefficient of $t$ is zero, the line is parallel to the plane (no intersection, or the line lies in the plane).

The direction $\mathbf{d}$ and normal $\mathbf{n}$ give the angle $\phi$ between line and plane via

Any point on the line and any scalar multiple of the direction works, so equivalent equations can look different.

Write the vector equation of the line through $(2, 1, 5)$ with direction $(1, 0, -2)$. [2 marks]

State a normal vector to the plane $3x - y + 4z = 7$. [1 mark]

Does the point $(1, 1, 1)$ lie on the plane $2x + y - z = 2$? [2 marks]