Matriks dan Ruang Vektor : Orthogonalitas, Proyeksi Vektor, dan Luas Jajar Genjang dengan Vektor
Orthogonal Vectors
Two nonzero vectors u and v in R^n are said to be orthogonal (perpendicular) if u⋅v = 0
Example: show that u = (-2,3,1,4) and v = (1,2,0,-1) are orthogonal vectors in R^4
Projection
u and a are vectors in R^n, and if a ≠ 0 :
vector component of u along a
vector component of u orthogonal to a
Example: let u=(2,-1,3) and a=(4,-1, 2). Find the vector component of u along a and the vector component of u orthogonal to a
In R^2 the distance D between the point P_0 (x_0,y_0) and the line ax+by+c=0 is
In R^3 the distance D between the point P_0 (x_0,y_0,z_0) and the plane ax+by+cz+d=0 is
Example : find the distance D between the point (1,-4,-3) and the plane 2x-3y+6z=-1
Area of Parallelogram
If u and v are vectors in 3-space, then ‖u×v‖ is equal to the area of the parallelogram determined by u and v
Example: Find the area of the triangle determined by the points P_1 (2,2,0), P_2 (-1,0,2), and P_3 (0,4,3)
Sumber
Slide MRV : Orthogonality of Vectors
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