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Engineering Calculus Notes 16

# Engineering Calculus Notes 16 - xy-plane “straight...

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4 CHAPTER 1. COORDINATES AND VECTORS P R Q x 1 x 2 x y 1 y 2 y Figure 1.3: Distance in the Plane In an oblique system, the formula becomes more complicated (Exercise 14 ). The rectangular coordinate scheme extends naturally to locating points in space. We again distinguish one point as the origin O , and draw a horizontal plane through O , on which we construct a rectangular coordinate system. We continue to call the coordinates in this plane x and y , and refer to the horizontal plane through the origin as the xy -plane . Now we draw a new z -axis vertically through O . A point P is located by ±rst ±nding the point P xy in the xy -plane that lies on the vertical line through P , then ±nding the signed “height” z of P above this point ( z is negative if P lies below the xy -plane): the rectangular coordinates of P are the three real numbers ( x,y,z ), where ( x,y ) are the coordinates of P xy in the rectangular system on the xy -plane. Equivalently, we can de±ne z as the number corresponding to the intersection of the z -axis with the horizontal plane through P , which we regard as obtained by moving the
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Unformatted text preview: xy-plane “straight up” (or down). Note the standing convention that, when we draw pictures of space, we regard the x-axis as pointing toward us (or slightly to our left) out of the page, the y-axis as pointing to the right in the page, and the z-axis as pointing up in the page (Figure 1.4 ). This leads to the identi±cation of the set R 3 of triples ( x,y,z ) of real numbers with the points of space, which we sometimes refer to as three dimensional space (or 3-space ). As in the plane, the distance between two points P ( x 1 ,y 1 ,z 1 ) and Q ( x 2 ,y 2 ,z 2 ) in R 3 can be calculated by applying Pythagoras’ Theorem to the right triangle PQR , where R ( x 2 ,y 2 ,z 1 ) shares its last coordinate with P and its other coordinates with Q . Details are left to you (Exercise 12 ); the resulting formula is | PQ | = r △ x 2 + △ y 2 + △ z 2 = r ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 . (1.2)...
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