Finding the Angle Between Two Intersecting Planes using the Dot Product of their Normal Vectors https://youtu.be/baRwDDtmwPY

In this video I show that we can determine the angle between two intersecting planes by computing the dot product of their normal or perpendicular vectors.

#math #science #education #geometry #STEM

Vector, Parametric, and Symmetric Equations of a Line https://youtu.be/ZeaSgJSWRuA

In this video I go over the the 3 main equations of a line in 3D: vector equation, parametric equations, and symmetric equations.

#math #science #education #STEM #geometry

Vector and Scalar Equations of a Plane https://youtu.be/DuSUIKAavcw

In this video I review the vector and scalar equations of a plane. The vector equation is the dot product set to equal zero between normal vector with the position vector of any two points on the plane. The scalar equation is the same as the vector equation except the dot product is evaluated, which produces a scalar (or number).

#math #science #education #STEM #geometry

Determining Parallel and Perpendicular Vectors and Planes via the Cross and Dot Products https://youtu.be/_MHe1u2tqv8

In this video I review how to determine whether two vectors or planes are parallel or perpendicular. Two vectors are parallel of their cross product is zero, and are perpendicular if their dot product is zero. The angle between two planes can be determining the angle between their normal vectors: if the angle is zero then the planes are parallel.

#math #geometry #science #education #STEM

Determining if 3 Points Lie on the Same Line Using the Cross Product https://youtu.be/S1khfWCF1u4

In this video I go over a method for determining if 3 points lie on the same line by using the cross product of any 2 position vectors formed from the points. If the cross product is zero then the points all lie on the same line.

#math #science #geometry #STEM #education

Four Points are on a Plane if the Scalar Triple Product is Zero https://youtu.be/wjLcU4VIGFE

In this video I illustrate a method for determining of 4 points lie on the same plane by checking if their scalar triple product (or volume of a parallelepiped) is zero. If it is zero then it means the volume of the parallelepiped spanned by the position vectors between the 4 points is zero, aka a flat plane.

#math #science #geometry #education #STEM

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