We work through an alternate solution to the same problem as in (1): Define a si

Question

We work through an alternate solution to the same problem as in (1): Define a single-variable function f[t) that is equal to the square of the distance from P [xo, yo, zo) to an arbitrary point on the line ' given by the vector-valued function r(t) = hat + x_1bt +y_1Ct + z_1: Use the method of optimization (from single-variable calculus) to find and justify the location of the global minimum in f(t): Use your formula (or follow the steps that derive the formula) to find the distance d from f 1, 2, 31 to the line given by r(t) = hi - t, 1 + t, 2ti.

Explanation / Answer

distance btwn 2 points (x1,y1,z1) and (x2,y2,z2) is given by sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2)

a)f(t) = (hat+x1-x0)^2+(bt+y1-y0)^2+(ct+z1i-z0)^2

b)derivative = 2(hat+x1-x0)ha+2(bt+y1-y0)b+2(ct+z1i-z0)c

derivative = 0 -->

t(h^2a^2+b^2+c^2) = hax0+y0b+z0c-hax1-y1b-z1ic

t = (hax0+y0b+z0c-hax1-y1b-z1ic)/(h^2a^2+b^2+c^2)

c)d = sqrt[(h1-t-1)^2+(t-1)^2+(2ti-3)^2]

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