mthsc206-fall-2010-notes-13.1

mthsc206-fall-2010-notes-13.1 - plane the distance formula...

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13 Vectors and the Geometry of Space 13.1 Three-Dimensional Coordinate System Tools: graphing in 1-dimension and 2-dimensions distance formula on the real number line distance formula in the plane Pythagorean Theorem completing the square Recall the coordinate systems for 1-dimensional (real number line) and 2-dimensional (the plane) spaces. We want to generalize to 3-dimensions and beyond. What does the coordinate system look like for a 3-dimensional space? Beyond? What is the right-hand rule? Where does this put the positive z -axis? Where would a left-hand rule place the positive z -axis? Describe the solutions to the equations x = 0, y = 0, z = 0, x = a , y = b , z = c . What would the solutions to the equation y = x look like? y = sin x ? x 2 + y 2 = 4? Problem 13.1.1 Let P 0 ( x 0 , y 0 , z 0 ) and P 1 ( x 1 , y 1 , z 1 ) be two points in 3-space. Determine the formula for the distance between P 0 and P 1 . Here are your only tools:
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Unformatted text preview: plane, the distance formula along any vertical or horizontal line, and the Pythagorean Theorem. Use these tools to derive the distance formula between the arbitrary points P and P 1 . P_0 P_1 Problem 13.1.2 Now that we have a distance formula for 3-space, what should the deFnition of a sphere with radius r > and center ( h, k, l ) be? Problem 13.1.3 Describe the sphere with equation 2 x 2 + 2 y 2 + 2 z 2 =-20 x + 12 y-4 z-52 . Problem 13.1.4 Describe the set of points that satisfy x 2 + y 2 + z 2 ≤ 4 . Problem 13.1.5 ±ind an equation of the set of points that are equidistant from the points A (1 , 2 , 3) and B (-1 , 4 , 7) . What does this set of points look like? 1...
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