Matriks dan Ruang Vektor : Norm, Dot Product, Cross Product, dan Panjang Vektor
Norm
‖v‖ : the norm of v, the length of v, or the magnitude of v. If v = (v_1,v_2,…,v_n) is a vector in R^n, then
Example : Suppose that v=(-3, 2, 1) in R^3, determine ‖v‖ !
Unit Vector
Definition of Unit Vector is a vector of norm 1. If v is any nonzero vector in R^n and u is a unit vector of v, then :
The process of multiplying a nonzero vector by the reciprocal of its length to obtain a unit vector is called normalizing v.
Example : Find the unit vector u that has the same direction as v = ( 2,2,-1 ) !
Distance
P_1 and P_2 are points in R^2 or R^3. The length of the vector (P_1 P_2 ) ⃗ is equal to the distance d between the two points. Specifically, if P_1 (x_1,y_1) and P_2 (x_2,y_2) are points in R^2, then
Similarly, the distance between the points P_1 (x_1,y_1,z_1 ) and P_2 (x_2,y_2,z_2) in 3-space is
Distance in R^n
u = (u_1,u_2,…,u_n) and v = (v_1,v_2,…,v_n) are points in R^n. The distance between u and v is d(u,v) :
Example : If u = (1, 3, -2, 7) and v = (0, 7, 2, 2), determine the distance between u and v!
Dot Product
u and v are nonzero vectors in R^2 or R^3. θ is the angle between u and v. Dot product of u and v is denoted by u⋅v
u(u_1,u_2,…,u_n) and v(v_1,v_2,…,v_n) are vectors in R^n. Dot product of u and v is denoted by u⋅v
Example: u = (2,1) and v = (-1,-2), determine u⋅v !
If θ is the angle between u and v, then
Example : determine the angle between u = (2,1) and v = (-1,-2) !
If u,v, and w are vectors in R^n, and if k is a scalar, then :
Cross Product
u = (u_1,u_2,u_3) and v = (v_1,v_2,v_3) are vectors in 3-space. The cross product u×v defined by
Cross product is constructing a vector in 3-space that is perpendicular to two given vectors. To determine the value of given Cross Product, just calculate or determine the determinant by using either Cofactor, Sarrus, or Row Reduction method.
Same as 3x3, 2x2 matrix vector also same method with 3x3 ones, but add 0 in third column of matrices and renamed it as k.
Example: Find u×v, where u = (1,2,-2) and v = (3,0,1) !
Relationships Involving Cross Product and Dot Product
Properties of Cross Product
Sumber
Slide MRV : Dot Product and Cross Product of Vectors
Posting Komentar