Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 48kijGeometric Properties of the Cross ProductThe following theorem shows that the cross product of two vectors is orthogonal to bothfactors.Theorem 3.11 If u and v are vectors in 3-space, then:(a) u·(u×v) = 0 (u×v is orthogonal to u)(b) v·(u×v) = 0 (u×v is orthogonal to v)Example 3.22 Let u and v be vectors as in Example 3.21. Show that u×v is orthogonalto both u and v.Solution .........Theorem 3.12 Let u and v be nonzero vectors in 3-space, and let θ be the angle betweenthese vectors when they are positioned so their initial points coincide.(a) ‖u×v‖ = ‖u‖‖v‖sinθ(b) The area A of the parallelogram that has u and v as adjacent sides isA = ‖u×v‖ (3.19)(c) u×v = 0 if and only if u and v are parallel vectors, that is, if and only if they arescalar multiples of one another.Example 3.23 Find the area of the parallelogram with two adjacent sides formed by thevectors u = 〈1,2,−2〉 and v = 〈3,0,1〉.Solution .........Example 3.24 Find the area of the triangle that is determined by the points P 1 (2,2,0),P 2 (−1,0,2), and P 3 (0,4,3).Solution .........