Curves and Surfaces - ME 210 Applied Mathematics for...

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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Equations of Lines and Planes in 3-D Space Straight Lines Consider the straight line, L. passing through a given point P 1 (x 1 , y 1 , z 1 ,) and parallel to a given vector . For any point, P(x, y, z) on the straight line:         1 1 1 1 PP = t V = x - x i + y - y j + z - z k = t a i + b j + c k  where t is any non-zero scalar. V = a i + b j + c k V 1 P P 1 1 1 x - x = t a y - y = t b z - z = t c are the parametric equations of the straight line, L. This can be written as 1 1 1 x - x y - y z - z = = t a b c
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 1 1 1 x - x y - y z - z = = = t a b c If anyone of the constants, a, b, and c, is zero, then the corresponding numerator must also be zero. The constants, a, b, and c are also the direction cosines of the straight line, i.e., the cosines of the angles between the line and the Cartesian coordinate axes, x, y, and z, respectively. It is also possible to represent a straight line as the line of intersection of two planes, i.e., a pair of plane equations together can be considered as the equation of the line of intersection between the planes.
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Find the equation of the straight line joining the points P(2, -1, 3) and Q(1, 0, -2). x y z O P Q PQ = - i + j - 5k M If M(x, y, z) is a point on the straight line, then QM = t PQ  The parametric equations of the line are x - 1 = t y = t z + 2 = - 5 t So, the equation of the line is z + 2 1 - x = y = - 5
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Do they intersect? If not, are they parallel or skew? Example: Consider the straight lines, L 1 and L 2 12 x - 1 y + 2 z - 3 x + 2 y + 1 z - 2 L : = = and = = 2 3 1 1 2 3 x - 1 = 2 t x + 2 = t y + 2 = 3 t y + 1 = 2 t z - 3 = t z - 2 = 3 t 2 t + 1 + 2 = t 3 t - 2 + 1 = 2 t t + 3 - 2 = 3 t 2 t - t = - 3 3 t - 2 t = 1 t - 3 t = - 1 These three equations with two unknowns are inconsistent. Therefore, there is no pair of values for t 1 and t 2 that satisfies all three equations. The lines are skew.
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ME – 210 Applied Mathematics for Mechanical Engineers Prof. Dr. Faruk Arınç Spring 2010 Example: Find the equation of the straight line, LL, passing through the point P(2, 3, -1) and (a) parallel to (b) perpendicular to the line L where x - 1 y + 1 z - 2 L : = = 2 3 1 x y z O L V (a) The given line L has the same direction as the vector V = 2 i + 3 j + k The equation of the straight line passing through P and parallel to L is x - 2 y - 3 = = z + 1 23
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Curves and Surfaces - ME 210 Applied Mathematics for...

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