Equations of planes (HL)

A plane, Π\Pi, is a flat surface, like an infinitely large square. It is the 2-D equivalent of a line. Two dimensional also means it requires two direction vectors to specify a plane.

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Using two directional vectors

r=d1λ+d2μ+(x0,y0,z0)\bm r = \bm{d_1}\lambda +\bm{d_2}\mu + (x_0, y_0, z_0)

where d1\bm{d_1} and d2\bm{d_2} are direction (displacement) vectors on the plane and are not scalar multiples of each other, and (x0,y0,z0)(x_0, y_0, z_0) is a point on the plane.

This roughly means that any displacement vector on the plane is a linear combination of d1\bm{d_1} and d2\bm{d_2}, meaning there exists a unique (λ,μ)(\lambda, \mu).

Normal vector

A normal vector n\bm n, or (a,b,c)(a, b, c), is one that is perpendicular to all displacement vectors on the plane. In particular, it can be found using the cross product of two direction vectors of the plane, meaning

k(a,b,c)=d1×d2k(a, b, c) = \bm{d_1} \times \bm{d_2}

where kk means the normal vector is any non-zero scalar multiple of the cross product of the two direction vectors.

Each displacement vector with respect to some point on the plane (x0,y0,z0)(x_0, y_0, z_0) can be represented by (xx0,yy0,zz0)(x - x_0,\,y - y_0,\, z - z_0)

Perpendicular vectors dot to 00, so

(xx0,yy0,zz0)(a,b,c)=0(x - x_0,\,y - y_0,\, z - z_0)\cdot(a, b, c) = 0

Vector equation of a plane

Rearranging the equation, we get

(x,y,z)(a,b,c)=(x0,y0,z0)(a,b,c)(x, y, z)\cdot(a, b, c) = (x_0, y_0, z_0)\cdot(a, b, c)

Scalar (Cartesian) equation of a plane

Simplifying the dot product on the right hand side, we get

ax+by+cz=dax + by + cz = d

To calculate distance between a plane and a point, it is useful to keep in mind dd is a dot product.

Note that unlike other forms, the Cartesian equation of a plane does not provide a point.

To find a point P(x0,y0,z0)P(x_0, y_0, z_0) on the plane, guess the component whose corresponding a,b,ca, b, c is not zero yet closest to zero. This ensures smallest denominator of fractions we have to work with. Then solve a system of two equations.

Because of the likelihood of working with fractions, you are very likely to show that a point is on the plane, or otherwise be given a point.

Wait, these are planes?

The xyxy-plane is represented by z=0z = 0. Similarly xzxz-plane is y=0y = 0, and yzyz-plane is x=0x = 0.

So in 2-D, x=0x = 0 is a line, but in 3-D it is a plane.

Normally, the plane ax+by+cz=dax + by + cz = d passes through the points (da,0,0){\left(\frac da, 0, 0\right)}, (0,db,0,){\left(0, \frac db, 0, \right)}, and (0,0,dc){\left(0, 0, \frac dc\right)} on the respective axes. However for the plane y=7y = 7, both aa and cc are 00, meaning y=7y = 7 intersects neither the xx-axis nor the zz-axis. Hence y=7y = 7 is parallel to the xzxz-plane.