Chapter 1 Topics in Analytic Geometry

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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 .........
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 49Scalar Triple ProductsIf u = 〈u 1 ,u 2 ,u 3 〉, v = 〈v 1 ,v 2 ,v 3 〉, and w = 〈w 1 ,w 2 ,w 3 〉 are vectors in 3-space, then thenumberu·(v×w)is called the scalar triple product of u, v, and w. This value can be obtained directlyfrom the formula∣ u 1 u 2 u 3∣∣∣∣∣u·(v×w) =v 1 v 2 v 3(3.20)∣w 1 w 2 w 3Example 3.25 Calculate the scalar triple product u·(v×w) of the vectorsSolution .........u = 3i−2j−5k, v = i+4j−4k, w = 3j+2kGeometric Properties of the Scalar Triple ProductTheorem 3.13 Let u, v and w be nonzero vectors in 3-space.(a) The volume V of the parallelepiped that has u, v and w as adjacent edges isV = |u·(v×w)| (3.21)(b) u·(v×w) = 0 if and only if u, v and w lie in the same plane.Example 3.26 Find the volume of the parallelepiped with three adjacent edges formed bythe vectors u = 〈7,8,0〉, v = 〈1,2,3〉 and w = 〈4,5,6〉.Solution .........Algebraic Properties of the Scalar Triple Productu·(v×w) = w·(u×v) = v·(w×u)u·v×w = u×v·w3.5 Parametric Equation of LinesLines Determined by a Point and a VectorA line in 2-space or 3-space can be determined uniquely by specifying a point on the lineand a nonzero vector parallel to the line.
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Chapter 1 Topics in Analytic Geometry

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