Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 36Rules of Vector ArithmeticTheorem 3.6 For any vectors u, v, and w and any scalars k and l, the following relationshipshold:(a) u+v = v+u(b) (u+v)+w = u+(v+w)(c) u+0 = 0+u = u(d) u+(−u) = 0(e) k(lu) = (kl)u(f) k(u+v) = ku+kv(g) (k +l)u = ku+lu(h) 1u = uNorm of a VectorThe distance between the initial and terminal points of a vector v is called the length, thenorm, or the magnitude of v and is denoted by ‖v‖. This distance does not change ifthe vector is translated. The norm of a vector v = 〈v 1 ,v 2 〉 in 2-space is given by‖v‖ =√v 2 1 +v 2 2and the norm of a vector v = 〈v 1 ,v 2 ,v 3 〉 in 3-space is given by‖v‖ =√v 2 1 +v2 2 +v2 3Example 3.7 Find the norm of v = 〈4,−2〉, and w = 〈−1,3,5〉.Solution .........is,For any vector v and scalar k, the length of kv must be |k| times the length of v; that‖kv‖ = |k|‖v‖Thus, for example,‖5v‖ = |5|‖v‖ = 5‖v‖‖−3v‖ = |−3|‖v‖ = 3‖v‖‖−v‖ = |−1|‖v‖ = ‖v‖This applies to vectors in 2-space and 3-space.Unit VectorsA vector of length 1 is called a unit vector. In an xy-coordinate system the unit vectorsalong the x-axis and y-axis are denoted by i and j, respectively; and in xyz-coordinate