Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 47The following cross products occur so frequently that it is helpful to be familiar withthem:i×j = k j×k = i k×i = j(3.18)j×i = −k k×j = −i i×k = −jThese results are easy to obtain; for example,i j k∣ ∣ ∣ ∣∣∣i×j =1 0 0∣0 1 0∣ = 0 0∣∣∣ 1 0∣ i− 1 0∣∣∣ 0 0∣ j+ 1 00 1∣ kHowever, rather than computing these cross products each time you need them, you canuse the diagram in Figure below.kijGeometric 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.