Distance from a Point to a Line using the Cross Product https://youtu.be/-ie6FTLTH0E

In this video I go over a method for determining the shortest distance between a point and a line in 3D by using the length of the cross product and the area of a parallelogram.

#math #science #geometry #education #STEM

Distance from a Point to a Plane using the Dot Product https://youtu.be/-V6F0OpS5M8

In this video I go over a quick recap on the formula for determining the shortest distance between a point and a plane in 3D. The formula is just the projection of any position vector from the plane to the point onto the normal vector of the plane, and involves their dot product.

#math #science #geometry #STEM #education

Distance Between Intersecting Lines, Parallel Lines, and Skew Lines https://youtu.be/hWqpXXNZK0U

In this video I discuss how to determine the distance between two lines for each of the 3 possible cases: intersecting lines, parallel lines, and skew lines (which don't intersect and are not parallel).

#math #science #geometry #education #STEM

Traces of a Surface: Review and Example on an Ellipsoid whose Traces are Ellipses https://youtu.be/w72pnqHHbAA

In this video I review traces of a surface, which are curves of intersection of a surface with planes parallel to the coordinate (x, y, z) planes.

#math #science #geometry #STEM #education

Equations of the Six Types of Quadric Surfaces https://youtu.be/0pCWT-FbmuY

In this video I review the standard form equations of the 6 types of quadric 3D surfaces: ellipsoid, cone, elliptic paraboloid, hyperboloid of one sheet, hyperbolic paraboloid, and hyperboloid of two sheets.

#math #science #geometry #education #STEM

Vectors and the Geometry of Space: Problems Plus https://youtu.be/4x6uh1zQ-M0 - notes - playlist

In this video I go over the end-of-chapter Problems Plus advanced math problems for the Vectors and Geometry of Space chapter. There are 8 problems in total and cover 3D geometry, surfaces, vectors, vector projections, and applications to physics and airplane dynamics. It's 4 hours of advanced math so buckle up!

#math #science #physics #geometry #calculus

Problems Plus 1: Tightly Packed Balls in a Box (and Tightly Packed Circles in a Square) https://youtu.be/Rs8YkurKjCM - notes - playlist

In this video I determine the radius of 9 balls that are tightly packed inside a box. I first solve the 2D case by solving for the radius of 5 circles tightly packed inside a square.

#math #science #education #geometry #STEM

Problems Plus 2: Volume Around a Box (and Area Around a Rectangle) https://youtu.be/bNZyuUB0D4w - notes - playlist

In this video I determine the volume of a region whose distance is at most 1 from a box. To understand how to solve this, I solve the 2D case first and solve the area around a rectangle.

#math #science #education #geometry #STEM

Problems Plus 3: Line and Surface of Two Intersecting Planes https://youtu.be/jj3wYCYhhq8 - notes - playlist

In this video I determine the symmetric equations of a line at the intersection of two planes. I then find out that when the equations of the plane vary, the line sweeps out a circular hyperboloid of one sheet. Finally, I calculate the volume of this surface between two planes.

#math #science #education #geometry #vectors

Problems Plus 5: Infinite Sum of Vector Projections is a Geometric Series https://youtu.be/rKt2qVLCMiM - notes - playlist

In this video I find the sum of an infinite series of consecutive magnitudes of vector projections and show it turns out to be just a geometric series.

#math #science #education #geometry #vectors

Problems Plus 6: Largest Sphere in a Sphere https://youtu.be/uefSE-rXFw4 - notes - playlist

In this video I find the largest sphere that passes through a given point while every point inside is also inside a larger sphere.

#math #science #geometry #education #GeoGebra

Problems Plus 7: The Physics of a Block on an Inclined Plane https://youtu.be/0ZJPy-Egfpk - notes - playlist

In this video I go over the physics of a solid block on an inclined plane in terms of the sliding friction, weight, normal force, and applied horizontal force.

#math #science #geometry #education #physics

Problems Plus 8: Volume of a Solid from its Shadows https://youtu.be/S-MGdEU9DF0 - notes - playlist

In this video I obtain the volume of the largest solid whose shadows are a square of height 1, a circle, and an isosceles triangle.

#math #science #geometry #education #physics

Vectors and the Geometry of Space: True-False Quiz https://youtu.be/8q0up5ccko8 - notes - playlist

In this video I go over the end-of-chapter True-False Quiz as a review for the Vectors and the Geometry of Space chapter. There are 22 questions in total covering vectors, cross products, dot products, lines, surfaces, and much more.

#math #science #physics #geometry #education

True-False Quiz Questions 1 to 4: False Statements about Vectors, Dot & Cross Products https://youtu.be/12t55GJHhxo - notes - playlist

In this video I go over the first 4 questions of the True-False Quiz on the Vectors and the Geometry of Space Chapter. These 4 questions are all false, and I discuss why for each question.

#math #science #physics #geometry #education

True-False Quiz Questions 5 to 7: Rearranging Vectors in the Dot and Cross Product https://youtu.be/bdRwfdjIeOE - notes - playlist

In this video I rearrange the vectors in the dot product and cross product and show that we get the same answer for the dot product but the negative value for the cross product.

#math #science #calculus #geometry #education

True-False Quiz Questions 8 and 9: Scalar Multiplication for Dot Product and Cross Product https://youtu.be/-YuGH9d41pw - notes - playlist

In this video I show that the dot product and cross product have a basic arithmetic multiplication property when multiplied by a scalar.

#math #science #calculus #geometry #education

True-False Quiz Questions 10 to 12: Rearranging Scalar and Vector Triple Products https://youtu.be/IHwBfOrJG_A - notes - playlist

In this video I show that interchanging the vectors in the scalar triple product result in the same answer but not when doing so for the vector triple product. I also show that adding vectors before taking a cross product is equivalent to taking two cross products and adding them up.

#math #science #calculus #geometry #education

True-False Quiz Questions 13 and 14: Dot and Cross Products with Repeating Vectors https://youtu.be/N2HVNiix8eI - notes - playlist

In this video I show two interesting formulas involving the dot and cross products in which a vector is used twice.

#math #science #STEM #geometry #education