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Engineering Calculus Notes 59

Engineering Calculus Notes 59 - Both are right angles...

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1.4. PROJECTION OF VECTORS; DOT PRODUCTS 47 B C A D E F G Figure 1.26: Euclid’s proof of Theorem 1.3.3 indefinitely, meet on that side on which are the angles less than the two right angles. Why do the interior angles between BC and the two angle bisectors add up to less than a right angle? ( Hint: What do you know about the angles of a triangle?) (b) Drop perpendiculars from D to each edge of the triangle, meeting the edges at E , F and G . (c) Show that the triangles △ BFD and △ BED are congruent. ( Hint: ASA—angle, side, angle!) (d) Similarly, show that the triangles △ CFD and △ CGD are congruent. (e) Use this to show that | DE | = | DF | = | DG | . (f) Now draw the line DA . Show that the triangles △ AGD and △ AED are congruent. ( Hint:
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Unformatted text preview: Both are right angles; compare one pair of legs and the hypotenuse.) (g) Conclude that ∠ EAD = ∠ ADG —which means that DA bisects ∠ A . Thus D lies on all three angle bisectors. 1.4 Projection of Vectors; Dot Products Suppose a weight is set on a ramp which is inclined θ radians from the horizontal (Figure 1.27 ). The gravitational force −→ g on the weight is directed downward, and some of this is countered by the structure holding up the ramp. The e²ective force on the weight can be found by expressing −→ g as a sum of two (vector) forces, −→ g ⊥ perpendicular to the ramp, and −→ g b...
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