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    10 Calculus Problems

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    Please see the attached file for the fully formatted problems.

    Problems involve:
    parametric equation of line segment,
    volume of a parallelipiped,
    sketching a plane given the equation,
    finding rectangular equations,
    center and radius of a sphere using the equation of a sphere,
    force vector problems.

    © BrainMass Inc. brainmass.com November 29, 2021, 11:54 pm ad1c9bdddf
    https://brainmass.com/math/basic-calculus/parametric-equation-line-segments-8537

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    1.)
    The important points are: intersection of plane with 3 axes x, y, z.
    x+2y+4z = 8
    Interception on x axis: y=0,z=0, x = 8
    Interception with y axis: x=0,z=0, y = 4
    Interception with zaxis: x=0,y=0, z = 2
    therefore,
    three well known points are: (8,0,0), (0,4,0), (0,0,2)
    See the attached file.

    2.)
    P(2,-1,4), Q(5,3,3)
    x1-x2 = 2 - 5 = -3
    y1-y2 = -1 - 3 = -4
    z1-z2 = 4 - 3 = 1
    therefore direction ratios of the line given as:
    a:b:c = -3:-4:1
    hence the parametric equation of straight line is given as:
    x = x1 + a*r
    y = y1 + b*r
    z = z1 + c*r
    => (x = 2 - 3r, y = -1 - 4r, z = 4 + r) --Answer

    3.)
    postion vectors of P,Q,R:
    P = (1,-1,2)=> vector OP = v(OP) = i - j + 2k
    Q = (4,1,3) => v(OQ) = 4i + j + 3k
    R = (-1,1,-1) => v(OR) = -i + j - k
    v(PQ) = v(OQ) - v(OP) = 3i + 2j + k
    v(RQ) = v(OQ) - v(OR) = 5i + 0.j + 4k
    direction cosines of the normal to the plane:
    normal vector to the plane = v(n)
    v(n) = v(PQ) X v(RQ) = 8i - 7j - 10k
    therefore, equation of the plane:
    {v(r) - v(OP)}.v(n) = 0
    because, r vector - OP vector will be in the plane and dot product ...

    Solution Summary

    The parametric equations of line segments are examined. A plane for an equation is sketched. Ten calculus problems are solved in detail.

    $2.49

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