Linear Help! Consider the e y mar in R2, and a point P (ro, yo) not lying on the

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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