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Engineering Calculus Notes 83

Engineering Calculus Notes 83 - A = 1 2 bh(1.24 where b is...

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1.6. CROSS PRODUCTS 71 are parametrizations of two distinct lines in P , both going through a point P 0 (with position vector −→ p 0 ), then −→ p ( s,t ) = −→ p 0 + s −→ v + t −→ w is a parametrization of the plane P . (b) Suppose an equation for P is Ax + By + Cz = D with C negationslash = 0. Show that the intersections of P with the xz -plane and yz -plane are given by z = A C x + D C z = B C y + D C and combine this with (a) to get a parametrization of P . (c) Apply this to the plane x + 2 y + 3 z = 9. 8. Find an equation for the plane P parametrized by x = 2 + s t y = 1 s + 2 t z = 3 + 2 s t. 1.6 Cross Products Oriented Areas in the Plane The standard formula for the area of a triangle
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Unformatted text preview: A = 1 2 bh, (1.24) where b is the “base” length and h is the “height”, is not always convenient to apply. Often we are presented with either the lengths of the three sides or the coordinates of the vertices (from which these lengths are easily calculated); in either case we can take a convenient side as the base , but calculating the height —the perpendicular distance from the base to the opposite vertex—can require some work....
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