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lines and planes

lines and planes - LINES and PLANES Maths21a O Knill LINES...

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LINES and PLANES Maths21a, O. Knill LINES. A point P and a vector vectorv define a line L . It is the set of points L = { P + tvectorv , where t is a real number } The line contains the point P and points into the direction vectorv . EXAMPLE. L = { ( x, y, z ) = (1 , 1 , 2) + t (2 , 4 , 6) } . This description is called the parametric equation for the line. EQUATIONS OF LINE. We can write ( x, y, z ) = (1 , 1 , 2) + t (2 , 4 , 6) so that x = 1 + 2 t, y = 1 + 4 t, z = 2 + 6 t . If we solve the first equation for t and plug it into the other equations, we get y = 1 + (2 x - 2) , z = 2 + 3(2 x - 2). We can therefore describe the line also as L = { ( x, y, z ) | y = 2 x - 1 , z = 6 x - 4 } SYMMETRIC EQUATION. The line vector r = P + tvectorv with P = ( x 0 , y 0 , z 0 ) and vectorv = ( a, b, c ) satisfies the symmetric equations x - x 0 a = y - y 0 b = z - z 0 c (every expression is equal to t ). PROBLEM. Find the equations for the line through the points P = (0 , 1 , 1) and Q = (2 , 3 , 4). SOLUTION. The parametric equations are ( x, y, z ) = (0 , 1 , 1) + t (2 , 2 , 3) or x = 2 t, y = 1 + 2 t, z = 1 + 3 t . Solving each equation for t gives the symmetric equations x/ 2 = ( y - 1) / 2 = ( z - 1) / 3. PLANES. A point P and two vectors vectorv, vectorw define a plane Σ. It is the set of points Σ = { P + tvectorv +
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