Linear Help! Consider the e y mar in R2, and a point P (ro, yo) not lying on the
ID: 2962315 • Letter: L
Question
Linear Help!
Consider the e y mar in R2, and a point P (ro, yo) not lying on the line. Find the point lin Q (z y") lying on the line, that is closest to P. Naturally, the coordinates of Q will depend on the values of m,zo, yo. Do this problem in two ways! 1. First, use "old fashioned' methods. Make a good sketch. Then, think hard about what geometric properties might allow you to find the coordinates of Q 2. Next, solve the problem using projections of vectors! (a) Rewrite the point P as a vector, say a. ind a vector b that is parallel to the line. (b Find the vector projection of a onto b 3. Does the vector you found in 2 match the point you found in 1? Was one of the methods easier than the other? 4. Think about how you would answer the similar problem of finding a point Q (z y*, z*) lying on a plane in R3, that is closest to some P (aco, yo, 20) lying off the plane. Describe some the difficulties you would face, and what you might need to overcome them!Explanation / Answer
1.
PQ is perpendicular to the line y = mx
=>
(y* -yo)/(x*-x0) = -1/m
=>
(mx*-y0)/(x*-x0) = -1/m
=>
m(mx*-y0)+(x*-x0) = 0
=>
x* = (my0+x0)/(m^2+1)
y* = m(my0+x0)/(m^2+1)
2.
(a)
a = x0 i +yo j
b = i+mj
(b)
projection of a onto b = (a.b)/|b|*(unit vector in the b direction) = (x0+my0)/(m^2+1)^0.5 *[i+mj]/(m^2+1)^0.5 = (my0+x0)/(m^2+1) i + m(my0+x0)/(m^2+1) j
3. the answers in both cases are same, it is easier the second way
4.
let equation of the plane be px+qy+rz = 0
a = xo i+ y0 j +z0 k
b = p i + q j +r k is the normal vector of the plane
=> unlike in above problem we cant use orthogonal projection here since the vector is not parallel but perpendicular to the plane.
let a1 bet the projection of a on b
=>
required vector = a-a1
=>
a1 = (px0 + qy0 + rz0)/(p^2+q^2+r^2)^0.5 *(pi + qj + rk)/(p^2+q^2+r^2)^0.5
= (px0+qy0+rz0)/(p^2+q^2+r^2)*p i + (px0+qy0+rz0)/(p^2+q^2+r^2)*q j + (px0+qy0+rz0)/(p^2+q^2+r^2)*r j
=>
a-a1=[x0 -(px0+qy0+rz0)*p/(p^2+q^2+r^2)] i +[y0- (px0+qy0+rz0)*q/(p^2+q^2+r^2)] j + [z0-(px0+qy0+rz0)*z0/(p^2+q^2+r^2)] k
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