Engineering Calculus Notes 65

# Engineering Calculus Notes 65 - cos 2 β cos 2 γ = 1(1.18...

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1.4. PROJECTION OF VECTORS; DOT PRODUCTS 53 and thus by Pythagoras’ Theorem | QR | 2 = | P 0 Q | 2 − | P 0 R | 2 = −→ w · −→ w p −→ w · −→ v | −→ v | P 2 = ( −→ w · −→ w )( −→ v · −→ v ) ( −→ w · −→ v ) 2 −→ v · −→ v . Another approach is outlined in Exercise 7 . Angle cosines: A natural way to try to specify the direction of a line through the origin is to Fnd the angles it makes with the three coordinate axes; these are sometimes referred to as the Euler angles of the line. In the plane, it is clear that the angle α between a line and the horizontal is complementary to the angle β between the line and the vertical. In space, the relation between the angles α , β and γ which a line makes with the positive x , y , and z -axes respectively is less obvious on purely geometric grounds. The relation cos 2 α
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Unformatted text preview: + cos 2 β + cos 2 γ = 1 (1.18) was implicit in the work of the eighteenth-century mathematicians Joseph Louis Lagrange (1736-1813) and Gaspard Monge (1746-1818), and explicitly stated by Leonard Euler (1707-1783) [ 4 , pp. 206-7]. Using vector ideas, it is almost obvious. Proof of Equation ( 1.18 ) . Let −→ u be a unit vector in the direction of the line. Then the angles between −→ u and the unit vectors along the three axes are −→ u · −→ ı = cos α −→ u · −→ = cos β −→ u · −→ k = cos γ from which it follows that −→ u = cos α −→ ı + cos β −→ + cos γ −→ k or in other words −→ u = (cos α, cos β, cos γ ) ....
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