For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line, \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) and a vector \(\mathbf{n}=\langle a, b, c\rangle\) that is parallel to the line. Then the equation of the line is \(x-x_{0}=a t, \ y-y_{0}=b t, \ z-z_{0}=c t\).)

\(x^{2}-8 x y z+y^{2}+6 z^{2}=0, P(1,1,1)\)

Text Transcription:

P_0_left(x_0,y_0,z_0_right)

mathbf_n=langle_a,b,c_rangle

x-x_0=at,y-y_0=bt,z-z_0=ct

x^2-8xyz+y^2+6z^2=0,P(1,1,1)

Force diagram A force diagram represents the forces acting on an object, and gives an indication of its motion, at a particular instant in time. Forces are represented with arrows that point toward (pushes) or away from (pulls) the object itself. The length of theses arrows is chosen to represent the relative strengths of each particular force (longer arrows mean stronger forces) and the direction of the arrows represents the direction the force is applied. Each force arrow is labeled to indicate how the force is exerted and its strength shown (in units of Newtons – N) when known. If the object in question is in motion at the moment at which the force diagram is drawn, this is indicated with a speed arrow; a half-arrow drawn close to the object (usually above for horizontal motion). The direction of the speed arrow represents the direction of motion and its length may be used to indicate relative speed, when comparisons of speeds (either of different objects, or different moments in time) are made. To determine what a shadow will look like on a screen you need to draw boundary rays: these are the two rays that leave the source and just pass by the outer edges of the blocker. Doing this will enable you to determine the size and position of the shadow region on the screen