Problem 1. - UCSD Mathematics

Problem the pointsP(1,1, 2),Q(2,0,1) andR(1, 1,0).(i) Find the area of the calculate~PQ= (1, 1,3),~RQ= (1,1,1).The area of the parallelogram spanned by~PQand~QRis the magnitude of the cross product(2, 1,3) (1,1,1) = ( 4,2,2).This vector has magnitude 2 6, so the trianglePQRhas area 6.(ii) Find the equation of the plane throughP, plane throughP, QandRhas as normal vector the cross product. The entries ofthe cross product are used as coefficients for the plane. We obtain the equation 4x+ 2y+ 2z= 6 2x+y+z= 3using the pointP(orQorR) to find the right hand 2.(i) Does there exist a constant such that the functionf(x,y,z) ={x4yx2+y2+z2if (x,y,z)6= (0,0,0)aif (x,y,z) = (0,0,0)is continuous?We must havea=lim(x,y,z) (0,0,0)x4yx2+y2+ that0 x4yx2+y2+z2 =x2x2+y2+z2 |x2y| |x2y| 0hencea= 0.(ii) Does the limitlim(x,y) (0,1)x2(y 1)2x4+ (y 1)4exist?We find the limit along the liney 1 =mx, keepingmfixed.}

Problem 3. Let ~x, ~y, and ~zbe vectors whose magnitudes are 1, 2, and 1 respectively. Suppose that ~xis parallel to (and in the same direction as) ~y, and ~xis perpendicular to ~z.

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