[Download] THE ARK OF MATHEMATICS PART 3: PROVING VECTORS AND VECTOR PRODUCTS by Chad Clawitter ~ Book PDF Kindle ePub Free

The ARK OF MATHEMATICS shows how to prove the fundamentals of Vectors for application in Engineering and Physics starting from the concrete definitions in Trigonometry and Geometry. The first lesson starts with understanding the concepts of vector addition, vector subtraction, and scalar multiplication, which are used to calculate the magnitude and components of a vector in 2 dimensions and 3 dimensions. The second lesson starts with the concrete definition of a “dot product” of two vectors using the idea of a projection of one vector onto another vector, which can be either be calculated using the magnitude of the vectors and the angle between or be calculated using the components of both vectors. It is shown how to calculate the dot product between two vectors in both 2 dimensions and 3 dimensions. The dot product of a vector with a unit vector in a specific direction is used to calculate the component of vector in that direction. The second lesson continues with the concrete definition of a “cross product” of two vectors being used to calculate a new vector that has magnitude equal to the area of the parallelogram spanned by the two vectors, which is pointing in a direction perpendicular to that parallelogram. It is shown how to calculate the cross product of two vectors, which can be either be calculated using the components of both vectors or by using the determinant of a square matrix filled with the unit vectors, components of the first vector, and components of the second vector. The second lesson finishes with a proof of the dot product identities, cross product identities, and proof of how to calculate the volume of a parallelepiped (a 3 dimensional generalization of a parallelogram) spanned by three vectors using the scalar triple product.