MAT223 Lecture 12

Completed Notes Status

Central objective: Apply vector operations (Dot Product, Cross Product, and Vector Projection) to solve geometric problems involving lines and planes in 3D space.
Key concepts:
- Vector Operations: Utilizing the cross product to find orthogonal vectors and the dot product as a component of vector projections. Links to Cross Product and Vector Projection.
- Plane Geometry: Extracting the Normal Vector directly from a plane's Cartesian equation coefficients to analyze intersections and parallel structures. Links to Normal Vector.
- Systems of Equations: Interpreting augmented matrices geometrically (e.g., 3 variables, 2 equations yielding 1 parameter implies planes intersecting in a line). Links to Intersection of Planes.
- Point to Plane Distance: Calculating the shortest distance using orthogonal projection of a test vector onto the plane's normal vector. Links to Point to Plane Distance.
Connections:
- Algebraic systems directly map to geometric configurations. The vector projection algebraically isolates the perpendicular component necessary for minimizing distances in geometric space.

Remember/Understand:
- How do you extract the normal vector from the plane equation $a x + b y + c z = d$ ?
- What is the geometric interpretation of a system with 3 variables and 2 equations?
Apply/Analyze:
- Given the vectors $\vec{x} = ⟨ 1, 1, 2 ⟩$ and $\vec{y} = ⟨ 1, - 1, 1 ⟩$ , calculate their cross product.
- Determine the shortest distance from the point $P (2, 1, - 3)$ to the plane $3 x - y + 4 z = 1$ .
Evaluate/Create:
- Formulate the equations of three distinct planes that intersect exactly at the origin $(0, 0, 0)$ .

Point to Plane Distance:
- Why it's challenging: It requires chaining several concepts: finding an arbitrary point on the plane, constructing a position vector, and projecting it onto the normal vector.
- Study strategy: Break the algorithm into three explicit steps: (1) Find $P_{0}$ , (2) Vector $\vec{P_{0} P}$ , (3) ${proj}_{\vec{n}} \vec{P_{0} P}$ . Practice executing these steps without referencing the formula sheet.
Skew Lines:
- Why it's challenging: Visualizing lines that do not intersect but are not parallel is unintuitive in 2D representations.
- Study strategy: Model the concept using physical pens in 3D space, mapping one to a specific plane and the other offset outside of it.

Immediate review actions:
- Review all [COMPLETED] insertions and verify with course materials
- Re-read Lecture Summary and confirm key links
Practice and application:
- Answer Remember/Understand questions
- Answer Apply/Analyze questions
- Attempt Evaluate/Create questions
Deep dive study:
- Focus on Challenging Concepts (above)
- Extend atomic notes only where the new lecture adds content
Verification and integration:
- Cross-reference notes with textbook/slides
- Identify remaining gaps
- Link this lecture to prior/future lecture notes