Title 83 Vector Calculus.pdf ✅

MSTC 82: Vector Calculus 1. Parts of Vectors A vector can be singly noted as a bold or italicized (or both) letter with a harpoon atop. When broken down into components, it can also be written in its component form, as shown: V⃑ = 〈Vx,Vy, . . . , Vz 〉 where x, y, . . . , z are the direction axes for the vector V⃑ . This notation is very useful when performing additive operations on multiple vectors. Another version of this notation is expressed in terms of unit vectors i,̂ĵ, . . . , k̂ along the direction axes such that V⃑ = Vxî+ Vyĵ, . . . , Vzk̂ Finally, the magnitude of the vector can be represented as ‖V⃑ ‖ or simply V. Components (Vi ) are projections of a vector along one direction axis from a set of axes, such that the sum of these projections results in the original vector. The ratio between a component of a vector and the vector itself is called a direction cosine. The term comes from the fact that the cosine of the angle subtended by the vector to one of its direction axes is equal to the ratio of the vector to the corresponding component. If the direction axes are perpendicular to each other, by trigonometry, cos θi = Ai ‖A⃑ ‖ Ai = ‖A⃑ ‖ cos θi where i is any of the direction axes (e.g. x, y, ... , z) for the vector. Finally, the relation between the magnitude and all of the components is V 2 = Vx 2 + Vy 2+. . . +Vz 2 Furthermore, a vector can also be represented using its unit vector λ, λV = 〈cos θx , cos θy , . . . , cos θz 〉 = V⃑ ‖V⃑ ‖ V⃑ = ‖V⃑ ‖λV Therefore, the unit vector is a representation of the original vector such that the components are its direction cosines, and the magnitude is equal to 1. A vector and its unit vector share the same line of action.


In terms of dot products, sab = ‖A⃑ ∙ B⃑⃑ ‖ ‖B⃑⃑ ‖ sba = ‖A⃑ ∙ B⃑⃑ ‖ ‖A⃑ ‖ 2.4.1. Vector Projection The vector projection of a vector A to another vector B is the vector with the magnitude of the scalar projection and the direction of the vector B. It can be computed using vab = sabλb vba = sbaλa In terms of dot products, vab = ‖A⃑ ∙ B⃑⃑ ‖ ‖B⃑⃑ ‖ B⃑⃑ ‖B⃑⃑ ‖ vba = ‖A⃑ ∙ B⃑⃑ ‖ ‖A⃑ ‖ A⃑ ‖A⃑ ‖ 2.4. Cross Product The cross-product multiplies two vectors to become a single vector perpendicular to both factors. A⃑ × B⃑⃑ = det [ î ĵ ... Ax Ay ... Bx By ... ] It is also equivalent to ‖A⃑ × B⃑⃑ ‖ = ‖A⃑ ‖‖B⃑⃑ ‖ sin θAB It can be done with a calculator using APPS. Vectors for the calculator are also selected from there. 3. Application of Vectors 3.1. Area of Triangles From Scissor’s formula, A = 1 2 ab sin θ If the sides a and b are expressed as vectors, then the area can be computed using A = 1 2 ‖a⃑ × b⃑ ‖ 3.2. Volume of Parallelepiped If the edges of the parallelepiped are expressed as vectors, then V = ‖a⃑ × b⃑ ‖ ∙ c⃑ Which edges are cross-multiplied can be interchanged. 3.3. Addition of Forces