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vb.net generate qr code If you multiply x by 1 and do not change the values of y or z, then point P will move parallel in Visual Studio .NET
If you multiply x by 1 and do not change the values of y or z, then point P will move parallel Decoding USS Code 39 In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Print Code 39 Full ASCII In Visual Studio .NET Using Barcode generator for .NET framework Control to generate, create USS Code 39 image in VS .NET applications. to the x axis to the opposite side of the yz plane, but P will end up at the same distance from the yz plane as it was before If you multiply y by 1 and do not change the values of x or z, then point P will move parallel to the y axis to the opposite side of the xz plane, but P will end up at the same distance from the xz plane as it was before If you multiply z by 1 and do not change the values of x or y, then point P will move parallel to the z axis to the opposite side of the xy plane, but P will end up at the same distance from the xy plane as it was before Reading Code39 In .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in .NET applications. Bar Code Drawer In Visual Studio .NET Using Barcode drawer for VS .NET Control to generate, create barcode image in VS .NET applications. Cartesian ThreeSpace
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Code39 Printer In .NET Framework Using Barcode generator for ASP.NET Control to generate, create Code 3/9 image in ASP.NET applications. Code 39 Full ASCII Drawer In VB.NET Using Barcode creator for .NET Control to generate, create Code 39 Extended image in .NET framework applications. In Cartesian threespace, the distance of a point from the origin depends on all three of the coordinates in the ordered triple representing the point The formula for this distance resembles the formula for the distance of a point from the origin in Cartesian twospace Bar Code Printer In VS .NET Using Barcode creator for .NET framework Control to generate, create bar code image in .NET framework applications. GS1 RSS Generator In .NET Using Barcode creator for .NET framework Control to generate, create GS1 DataBar Truncated image in VS .NET applications. The general formula It s not difficult to derive a general formula for the distance of a point from the origin in Cartesian threespace, as long as we re willing to use our spatial mind s eye Suppose we name the point P, and assign it the coordinates Create Barcode In VS .NET Using Barcode generator for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. Code 93 Full ASCII Maker In VS .NET Using Barcode maker for .NET framework Control to generate, create Code 93 Full ASCII image in Visual Studio .NET applications. P = (xp,yp,zp) Figure 74A shows this situation, along with a point P* = (xp,yp,0), which is the projection of P onto the xy plane We ve moved again back to the perspective of Fig 72, looking in toward the origin from somewhere far out in space near the negative y axis To find the distance of P* from the origin, we can work entirely in the xy plane This gives us a twodimensional distance problem, which we learned how to handle in Chap 1 Let s call the distance of P* from the origin by the name a Using the formula we learned in Chap 1 for the distance of a point from the origin in Cartesian xy plane, we have a = (xp2 + yp2)1/2 Read Barcode In Visual Studio .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET applications. Encoding EAN13 In ObjectiveC Using Barcode encoder for iPad Control to generate, create European Article Number 13 image in iPad applications. First, we find the distance from the origin to P*, and call it a
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Distance of Point from Origin
This completes the first step in a threephase process Figure 74B shows the second step Here, we find the distance between P* and P Let s call that distance b It s the perpendicular distance of P from the xy plane, which is simply the coordinate value zp Therefore, we have b = zp That s the end of the second step In Fig 74C, the distance from the origin to P is labeled c Note that we now have a right triangle with sides of lengths a, b, and c The right angle is between the sides whose lengths are a and b The Pythagorean theorem therefore allows us to make the claim that a2 + b2 = c2 Substituting the previously determined values for a and b into this formula gives us [(xp2 + yp2)1/2]2 + zp2 = c2 which simplifies to xp2 + yp2 + zp2 = c2 Second, we find the distance from P* to P, and call it b
P* (xp, yp, 0) x a
z P (xp, yp, zp) Figure 74B Finding the distance of point P from the origin: step 2
Cartesian ThreeSpace
Third, we call the distance from the origin to P by the name c +z
P* (xp, yp, 0) x Right angle b a c
z P (xp, yp, zp) and note that c is the length of the hypotenuse of a right triangle! Figure 74C Finding the distance of point P from the origin: step 3
When we switch the righthand and lefthand sides of this equation and then take the 1/2 power of both sides, we get the formula we ve been looking for, which is c = (xp2 + yp2 + zp2)1/2 An example Let s find the distance from the origin to the point P = ( 5, 4,3) as shown in Fig 73 We have xp = 5, yp = 4, and zp = 3 If we call the distance c, then c = (xp2 + yp2 + zp2)1/2 = [( 5)2 + ( 4)2 + 32]1/2 = (25 + 16 + 9)1/2 = 501/2 Another example Now let s find the distance from the origin to Q = (3,5, 2) as shown in Fig 73 This time, the coordinates are xq = 3, yq = 5, and zq = 2 We can again call the distance c, so c = (xq2 + yq2 + zq2)1/2 = [32 + 52 + ( 2)2]1/2 = (9 + 25 + 4)1/2 = 381/2

