91Ó°ÊÓ

Suppose that \(A, B,\) and \(C\) are vertices of a triangle and that \(a, b\) , and \(c\) are, respectively, the midpoints of the opposite sides. Show that \(\overrightarrow{A a}+\overline{B b}+\overline{C c}=0\)

Find the center \(C\) and the radius \(a\) for the spheres in Exercises \(51-60\) $$3 x^{2}+3 y^{2}+3 z^{2}+2 y-2 z=9$$

Unit vectors in the plane Show that a unit vector in the plane can be expressed as \(\mathbf{u}=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j},\) obtained by rotating \(\mathbf{i}\) through an angle \(\theta\) in the counterclockwise direction. Explain why this form gives every unit vector in the plane.

find the point in which the line meets the plane. \begin{equation}x=2, \quad y=3+2 t, \quad z=-2-2 t ; \quad 6 x+3 y-4 z=-12\end{equation}

Use a CAS to plot the surfaces in Exercises \(53-58 .\) Identify the type of quadric surface from your graph. $$ y-\sqrt{4-z^{2}}=0 $$

find the point in which the line meets the plane. \begin{equation}x=1+2 t, \quad y=1+5 t, \quad z=3 t ; \quad x+y+z=2\end{equation}

Consider a triangle whose vertices are \(A(2,-3,4), B(1,0,-1),\) and \(C(3,1,2) .\) \begin{equation}a. \overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A} . \quad \text { b. Find } \overrightarrow{B A}+\overrightarrow{A C}+\overrightarrow{C B}\end{equation}

Find the center \(C\) and the radius \(a\) for the spheres in Exercises \(51-60\) $$x^{2}+y^{2}+z^{2}-4 x+6 y-10 z=11$$

Find the center \(C\) and the radius \(a\) for the spheres in Exercises \(51-60\) $$(x-1)^{2}+(y-2)^{2}+(z+1)^{2}=103+2 x+4 y-2 z$$

find the point in which the line meets the plane. \begin{equation}x=-1+3 t, \quad y=-2, \quad z=5 t ; \quad 2 x-3 z=7\end{equation}