Let u and v be non-parallel unit vectors. Show that the vector (u + v) bisects t
ID: 2894534 • Letter: L
Question
Let u and v be non-parallel unit vectors. Show that the vector (u + v) bisects the angle between u and v. Write the equation of the line that is perpendicular to the plane 3x - y + z = -9 and passing through the point (1.-2, 1). Consider helix r(t) = (cos t, sin t,t), t elementof R. Show that r(t) is not planar. That is, there is no plane in R^3 that contains r(t). To show this observe that if a curve lies on a plane, the normal of the plane is always orthogonal to tangent vector of the curve at any point. Use this idea to obtain a contradiction. Write the equation of the plane containing three points (1,1,2).(-1,3,0) and (0,1,-4). Write the equation of the tangent line to the curve r(t) = (t^2, 2-t^3,e^t) at the point (1, 1, e). Write the equation of the plane that contains the line (2 + t, -t, -1 + 2t) and the point (1, 1,-5).Explanation / Answer
8)
let the points on the plane are defined as A(1,1,2),B(-1,3,0),C(0,1,-4),X(x,y,z)
AX=X-A=<x-1,y-1,z-2>
AB=B-A=<-1-1,3-1,0-2> =<-2,2,-2>
AC=C-A=<0-1,1-1,-4-2> =<-1,0,-6>
ABxAC=<(2*-6)-(0*-2),(-1*-2)-(-2*-6),(-2*0)-(-1*2)>
ABxAC=<-12,-10,2>
(ABxAC).AX=0
<-12,-10,2>.<x-1,y-1,z-2>=0
=> -12(x-1)-10(y-1) +2(z-2) =0
=>-12x +12 -10y +10 +2z -4=0
=>-12x-10y+2z=-18
=>6x+5y-z=9
equation of plane containing three points (1,1,2),(-1,3,0),(0,1,-4) is 6x+5y-z=9
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