Find two different planes whose intersection is the line x=1+t, y=2-t, z=3+2 t . Write equations

Find two different planes whose intersection is the line...

Key Concepts

Parametric Representation of a Line

This concept involves expressing the coordinates of a line as functions of a parameter, typically denoted as t. It defines a point on the line and a direction vector, allowing for a concise description of all points along the line in three-dimensional space.

Plane Equation in Three-Dimensional Space

A plane in three-dimensional space is commonly described by a linear equation of the form Ax + By + Cz = D. This equation utilizes a normal vector—whose components are A, B, and C—which is perpendicular to the plane, and a scalar D that adjusts the plane's position relative to the origin.

Intersection of Two Planes

When two non-parallel planes in three-dimensional space intersect, their intersection is a line. This concept explores how the geometric properties of the planes, particularly their normal vectors and positions, result in a one-dimensional intersection where every point satisfies both plane equations.

Normal Vectors

A normal vector is a vector that is perpendicular to a given plane. It plays a crucial role in defining the orientation of a plane and forms the basis for writing the plane's equation. The components of the normal vector directly correspond to the coefficients in the plane’s linear equation, encapsulating the plane's directional properties.

Transcript

00:01 We're looking to find two planes that share a line of intersection given by x equals 1 plus t, y equals 2 minus t, and z equals 3 plus 2 t.

00:14 So i'm going to write this equation of the line as 1, 2, 3 plus t times 1 minus 1.

00:32 If you think about what we have, we have a plane, and then we have some other plane here, and this plane has this line of intersection given by this equation.

00:48 But this line could also be the line of intersection for this plane, or it could be the line of intersection for this plane that comes to here, or an infinite number of planes.

01:00 So the assignment of the equations for these two planes is somewhat arbitrated.

01:05 What we do know, however, is that if i have this plane and this is the line on the plane, this direction vector, i'll call you, is this direction vector here, 1 minus 1, 2.

01:27 There exists a normal to this plane that is perpendicular to this line perpendicular to this plane.

01:40 This will have an arbitrary assignment abc because we don't know what the normal to this plane is because we don't know what this plane is.

01:50 What we do know is that this normal is perpendicular to the plane and therefore it is perpendicular to this direction vector.

02:00 And if they are perpendicular, then we know that the dot product of u and n must equal zero.

02:09 That means that this dot product 1 times a plus negative 1 times b plus 2 times c must equal 0.

02:25 So at this point, we just arbitrarily choose an a and a b and a c that satisfy this equation.

02:31 If we do it, it will be a normal.

02:37 Doing that, i have 1 times let's choose 2.

02:44 And negative 1, i'll choose 4.

02:49 So now i have 2 minus 4 is negative 2.

02:52 That means i would have to add two more to that to make it equal 0.

02:58 So i have the a, b, and c here as the vector, the normal vector, 2, 4, and 1.

03:17 If this normal vector is 2, 4 and 1, if you recall the equation of a plane is ax plus by plus cz is equal to d.

03:28 And these a, bs, and cs come from the normal vector, or shared with the normal vector, a .i plus bj plus ck.

03:39 So one plane has got coefficients of 2, 4, and 1...

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