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23 June, 13:39

x + y + z = 2, x + 3y + 3z = 2 (a) Find parametric equations for the line of intersection of the planes. (Use the parameter t.) (x (t), y (t), z (t)) = (b) Find the angle between the planes. (Round your answer to one decimal place.)

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  1. 23 June, 14:35
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    a) L (t) = (2,0,0) + (0,-1,1) * t

    b) θ = 0.384 rad

    Step-by-step explanation:

    a) for the planes

    x + y + z = 2

    x + 3y + 3z = 2

    then substracting the first equation to the second

    2y+2z=0 → y = - z

    replacing in the first equation

    x + (-z) + z=2 → x=2

    thus choosing t=z as parameter

    x=2

    y=-t

    z=t

    or

    L (t) = (2,0,0) + (0,-1,1) * t

    b) the angle can be found through the dot product of the normal vectors to the plane

    n1*n2 = (1,1,1) * (1,3,3) = 1*1+1*3+1*3 = 7

    |n1| = √ (1²+1²+1²) = √3

    |n2| = √ (1²+3²+3²) = √19

    since

    n1*n2 = |n1|*|n2|*cos θ

    cos θ = n1*n2 / |n1|*|n2| = 7 / (√3*√19) = 0.927

    thus

    θ = cos⁻¹ 0.927 = 0.384
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