Engineering Calculus Notes 26

Engineering Calculus Notes 26 - 1 and Q x 2,y 2,z 2 let R...

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14 CHAPTER 1. COORDINATES AND VECTORS 10. What conditions on the spherical coordinates of a point signify that it lies on (a) the x -axis? (b) the y -axis? (c) the z -axis? (d) the xy -plane? (e) the xz -plane? (f) the yz -plane? 11. A disc in space lies over the region x 2 + y 2 a 2 , and the highest point on the disc has z = b . If P ( x,y,z ) is a point of the disc, show that it has cylindrical coordinates satisfying 0 r a 0 θ 2 π z b. Theory problems: 12. Prove the distance formula for R 3 (Equation ( 1.2 )) | PQ | = r x 2 + y 2 + z 2 = r ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 + ( z 2 z 1 ) 2 . as follows (see Figure 1.9 ). Given P ( x 1 ,y 1 ,z
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Unformatted text preview: 1 ) and Q ( x 2 ,y 2 ,z 2 ), let R be the point which shares its last coordinate with P and its ±rst two coordinates with Q . Use the distance formula in R 2 (Equation ( 1.1 )) to show that dist( P,R ) = r ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 , and then consider the triangle △ PRQ . Show that the angle at R is a right angle, and hence by Pythagoras’ Theorem again, | PQ | = R | PR | 2 + | RQ | 2 = r ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 ....
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