Find an equation of the plane. The plane that passes through the point (3,5,-1) and contains the

Find an equation of the plane. The plane that passes through...

Key Concepts

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in another vector perpendicular to both. This property is especially useful in finding a normal vector to a plane when two independent, non-parallel vectors lying on the plane are known.

Plane Equation

The plane equation is a mathematical expression that defines a flat, two-dimensional surface in three-dimensional space. It is usually expressed in the form Ax + By + Cz + D = 0, where (A, B, C) is the normal vector to the plane, representing the direction perpendicular to the plane, and D is a constant that positions the plane relative to the origin.

Point on a Plane

A point on a plane is any given coordinate through which the plane passes. In determining the plane equation, having at least one point on the plane gives a specific reference that, combined with the normal vector, allows for the complete determination of the plane's positioning in space.

Parametric Equations of a Line

Parametric equations express the coordinates of the points on a line as functions of a single parameter. This representation is useful because it provides both a specific point on the line and a direction vector, which can be used in problems involving intersections and eliminations of parameters.

Direction Vector

A direction vector specifies the orientation or direction in which a line extends in three-dimensional space. In the context of lines and planes, the direction vector is essential in determining parallelism or in forming a basis for the plane when combined with another vector.

Normal Vector

The normal vector of a plane is a vector that is perpendicular to every line that lies on the plane. It is a critical component for writing the plane equation, as its components serve as the coefficients of x, y, and z in the equation Ax + By + Cz + D = 0.

Transcript

00:01 For this question, we are required to compute the equation for the plane, let's call it p, which passes through the point a and contains a given straight line l.

00:15 Okay, if we write the equation for the straight line in detail, we can write it as 4 minus 0 plus t times minus 1 to minus 3.

00:31 Okay, so we know l is a straight line which passes through this point, let's call it b, 4 minus 1, 0, with a directional vector minus 1 to minus 3.

00:47 Okay, let's call it v, v1.

00:52 As p contains a straight line, that means the vector v1 must be parallel to our plane p, right, because the straight line is in this plane, that means the vector must be parallel to a given plane.

01:10 Okay, now combine the condition that the straight line is in this plane, that means the point b is also in our plane.

01:22 Okay, so now let's define something new.

01:25 Define the vector 2 to be just equal to the vector from a to b.

01:32 Okay, as both a and b are contained in our plane p, we know this vector must also be parallel to our plane p, right, because that means once those two points are both contained in our plane, the straight line determined by those two points is also contained in this plane p.

01:59 And the direction vector, which is just vab, must also be parallel to this plane.

02:08 Okay, now what is the coordinate for this vector? by the coordinate of a and b, it's easy for us to know this vector will be actually equal to 1 minus 6, 1.

02:27 Okay, and now assume n is a vector normal to our plane p, and assume the coordinate for n is x, y, z.

02:49 We know as v1, v2 are both parallel to p, we know the inner product of n with v1 will be equal to the inner product between n and v2 is equal to 0.

03:05 Right, now by the first equation, we have minus 1 minus x plus 2y plus minus 3z is equal to 0.

03:19 This is the first equation...

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