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Sec10.5 - Sec 10.5 Equation of Lines and Planes In 2D a...

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Sec. 10.5 Equation of Lines and Planes In 2D a line is determined by a point and a slope (direction). In 3D a line L is determined by a point 0 0 0 0 ( , , ) P x y z = and the direction of L which is given by a vector v that is parallel to L . The vector equation of line L: 0 r r tv = + (Putting bounds on t will yield a line segment.) As t varies the line is traced out by the tip of vector r. (v is the directional vector of the line.) Let v = < a, b, c > Æ tv = < at, bt, ct >. Let r = < x, y, z > and 0 0 0 0 , , r x y z =< > . Substituting into the vector equation of the line we can get the component form. The component form of the vector equation of line L: 0 0 0 , , r x at y bt z ct =< + + + > Since two vectors are equal if and only if their components are equal, we get the parametric form of a line in 3 space where t is a real number. The parametric form of line L: 0 0 0 x x at y y bt z z ct = + = + = + through 0 0 0 0 ( , , ) P x y z = and parallel to v = < a, b, c >. Note: The vector form and the parametric form are not unique!

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Sec10.5 - Sec 10.5 Equation of Lines and Planes In 2D a...

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