Picture a flat tabletop with a pencil standing straight up: A plane is a flat sheet that goes on forever. To pin it down you need one point on it and a direction that points straight out of it — that perpendicular direction is the normal vector n (think of a pencil balanced upright on a tabletop).
If a is the position vector of a known point on the plane and r is the position vector of any other point on it, then r − a lies flat in the plane, so it is perpendicular to n. Two perpendicular vectors have scalar product zero, so (r − a)·n = 0, which rearranges to the form below.
IB-style question — write the scalar-product form
A plane passes through the point A(1, 2, −1) and has normal vector n = (3, −1, 2).
Write the equation of the plane in the form r·n = a·n.
Step by step
- The normal n goes straight into the equation on the left.
- Work out the right-hand number a·n using the known point A(1, 2, −1).
- Put the number in.
Final answer
r·(3, −1, 2) = −1.
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Expand the dot product and the coefficients ARE the normal: Write r = (x, y, z) and let n = (a, b, c). Expanding r·n = d gives the Cartesian form:
ax + by + cz = d.
The huge payoff: the coefficients of x, y, z are exactly the normal vector. So you can read the normal straight off any Cartesian plane equation — no work needed.
IB-style question — convert to Cartesian form
A plane has normal n = (4, −2, 5) and passes through B(2, 0, 3).
Find the Cartesian equation of the plane.
Step by step
- Set up ax + by + cz = d with the normal as the coefficients.
- Find d by substituting the point B(2, 0, 3) the plane passes through.
- State the equation.
Final answer
4x − 2y + 5z = 23.
IB-style question — is the point on the plane?
Determine whether the point P(3, 1, 2) lies on the plane 4x − 2y + 5z = 23.
Step by step
- Substitute P's coordinates into the left-hand side.
- Compare with the right-hand side (23). Since 20 ≠ 23, the equation is not satisfied.
Final answer
No — P does not lie on the plane (the equation is not satisfied).