(a) Explain why the line of intersection of two planes must...

(a) Explain why the line of intersection of two planes must be parallel to the cross product of a normal vector to the first plane and a normal vector to the second. (b) Find a vector parallel to the line of intersection of the two planes $x+2 y-3 z=7$ and $3 x-y+z=0$ (c) Find parametric equations for the line in part (b).

(a) Explain why the line of intersection of two planes must be parallel to the cross product of a normal vector to the first plane and a normal vector to the second.
(b) Find a vector parallel to the line of intersection of the two planes $x+2 y-3 z=7$ and $3 x-y+z=0$
(c) Find parametric equations for the line in part (b).

Key Concepts

Plane Definition and Normal Vectors

A plane in three-dimensional space is given by an equation of the form Ax + By + Cz = D, where the coefficients (A, B, C) form a normal vector to the plane. This normal vector is crucial because it is perpendicular to every direction within the plane, thereby characterizing the orientation of the plane.

Cross Product

The cross product of two vectors results in a new vector that is perpendicular to both of the original vectors. When applied to the normals of two planes, the resulting cross product is parallel to the line of intersection of the planes, since that line is the common direction that is orthogonal to both normals.

Parametric Equations of a Line

A line in space can be described parametrically by a point on the line and a direction vector. The equations take the form x = x0 + at, y = y0 + bt, and z = z0 + ct, where (x0, y0, z0) is a specific point on the line, (a, b, c) is a vector parallel to the line, and t is a scalar parameter that ranges over all real numbers.

Finding the Intersection of Two Planes

When two non-parallel planes intersect, their intersection is a line. The line’s direction can be determined by taking the cross product of the normal vectors of the two planes, as this gives a vector that is perpendicular to both normals. Once the direction vector is known, finding any point that satisfies both plane equations allows for the formulation of the line’s parametric equations.

Transcript

00:01 So we are given with that proof or argument that the line of intersection between two planes has to be parallel to the cross product of a normal vector to the first and normal vector to the second.

00:10 Let's go ahead and draw some planes here.

00:15 Imagine this is a plane and this is a plane and they're intersecting along this line.

00:22 Let's actually draw that even in there.

00:27 So you must be parallel to the cross product of a normal vector to the first plane.

00:31 Let's draw that like that and normal vector to the second.

00:35 Well, let's see.

00:37 So the cross product of two normal of two vectors is going to be perpendicular to both of those vectors.

00:45 So the cross product of this with anything will be perpendicular to...

00:51 If we give it...

00:52 Let's give these planes some names.

00:54 So this plane is plane one, plane two.

00:58 So this can be normal vector one, vector two.

01:03 We're saying that normal vector one cross any vector v is going to be perpendicular to normal vector one, which in particular means that it is parallel to plane one, right? because normal vector one is defined to be parallel...

01:23 Perpendicular to plane one.

01:25 Likewise, normal vector two cross product with any v must be parallel to plane two and as such will be as such parallel to plane two.

01:43 So together normal one cross normal two must be parallel to plane one and parallel to plane two, which is to say that it's that it is on...

01:59 It's parallel to their line of intersection as well.

02:04 So the line of intersection is then parallel to the cross product of both of those things by basically transitivity of being parallel.

02:12 That's great, that's cool...

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