Annotated-Emath_Week-2_Vec_Geom

Checkpoints

• Regular Updates:

  • Check your course emails or the Learn News forum regularly.

  • Review tutorial solutions weekly (available from 5 PM on Thursdays).

  • Be aware of work due each week:

    • Tutorial homework questions by your tutorial time.

    • Quiz due at 5 PM on Fridays.

• Topics This Week:

  • Planes

  • Cross products

  • Finding areas and volumes using vectors

  • Solving intersection and distance problems

Planes in 3-Space

• Vector Representation of a Plane:

  • A plane in 3-space can be fixed by:

    • A position vector (r_0) and a direction vector (d)

    • Two independent vectors can be formed by subtracting two points.

  • Line Representation:

    • ( \mathbf{r} = (x, y, z) ) where (s \in R) is an arbitrary parameter, represented as:

      • (\mathbf{r} = \mathbf{r_0} + s\mathbf{d})

• Parametric Equation of a Plane:

  • A plane is 2-dimensional and requires two parameters.

  • General form: ( \mathbf{r} = \mathbf{r_0} + s\mathbf{d} + t\mathbf{e} )

    • Where (s, t \in R) and (\mathbf{d}) and (\mathbf{e}) are direction vectors parallel to the plane.

Defining a Plane in R³

• Ways to Define a Plane:

  1. One point on the plane and two independent vectors parallel to the plane.

  2. Three non-collinear points on the plane.

  3. One point on the plane and one vector perpendicular (normal) to the plane.

  4. One consistent linear equation in three unknowns (scalar equation):

     • (ax + by + cz = d)

• Examples of Point and Vector Definitions:

  • (\mathbf{r} = \mathbf{r_0} + s\mathbf{d} + t\mathbf{e}) yields the vector parametric form of the plane.

  • The linear combination (s\mathbf{d} + t\mathbf{e}) results in any point on the plane, where (\mathbf{r_0}) is a specific point on it.

Using Three Points to Define a Plane

• Suppose points A, B, and C are given by position vectors:

  • Take (\mathbf{r_0} = \mathbf{a}) and choose direction vectors (\mathbf{d} = \mathbf{b-a}) and (\mathbf{e} = \mathbf{c-a}).

  • This gives the parametric representation:

    • (\mathbf{r} = \mathbf{r_0} + s\mathbf{d} + t\mathbf{e})

Exercise 31: Determining a Plane

• Find the plane through the points (1, 2, 1), (5, 0, −1), and (3, −1, −1).

  • Use (\mathbf{r_0} = (1, 2, 1)).

  • Direction vectors:

    • (\mathbf{d} = (4, -2, -2))

    • (\mathbf{e} = (2, -3, -2))

  • Resulting vector parametric equation:

    • (\mathbf{r} = (1, 2, 1) + s(4, -2, -2) + t(2, -3, -2))

Point-Normal Form of a Plane

• A plane can also be represented as:

  • ((\mathbf{r} - \mathbf{r_0}) \cdot \mathbf{n} = 0)

    • Where (\mathbf{n}) is the normal vector.

    • This is the point-normal form.

    • If a point (\mathbf{x}) lies on the plane, then ((\mathbf{x}-\mathbf{r_0}) \cdot \mathbf{n} = 0).

Scalar Form of a Plane

• An equation of the form (ax + by + cz = d) describes a plane in 3-space with normal vector (\mathbf{n} = (a, b, c)).

• At least one of (a, b, c) must be non-zero to represent a valid plane.

Exercises: Form Transformations

1. Convert Scalar to Vector Parametric Form:

   • Given equation format, introduce parameters and rearrange.

2. Convert Vector Point-Normal to Scalar Form:

   • Expand and derive coefficients for the plane equation from the point-normal format.

Cross Product Fundamentals

• Definition: The cross product of two vectors results in a vector orthogonal to both.

• Formula: For vectors (\mathbf{x}) and (\mathbf{y}):

  • (\mathbf{x} \times \mathbf{y} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \mathbf{x}_1 & \mathbf{x}_2 & \mathbf{x}_3 \ \mathbf{y}_1 & \mathbf{y}_2 & \mathbf{y}_3 begin{matrix} \end{vmatrix} )

• Direction of the Cross Product:

  • Determined by the right-hand rule.

• Properties:

  • Non-commutative (i.e., (\mathbf{x} \times \mathbf{y} = - (\mathbf{y} \times \mathbf{x}))).

Areas and Volumes with Cross Products

• Area of Parallelogram: = ( ||\mathbf{x} \times \mathbf{y}|| )

• Area of Triangle: = ( \frac{1}{2} ||\mathbf{x} \times \mathbf{y}|| )

• Volume of Parallelepiped: = ( |\mathbf{x} \cdot (\mathbf{y} \times \mathbf{z})| )

Intersection Problems

• Lines Meeting Planes:

  • To find the intersection of a line and a plane, ensure the plane's equation is in scalar point-normal form and solve for the parameter.

  • Confirm intersection by substituting back into the line's equation to find coordinates.

• Finding Distances:

  • Distance from a point to a plane involves determining the line orthogonal to the plane and solving for intersection coordinates to compute the distance.