91Ó°ÊÓ

Find the unit tangent vector for the curve having the given vector equation. \(\mathbf{R}(t)=\sin 2 t \mathbf{i}+\cos 2 t \mathbf{j}+2 t^{3 / 2} \mathbf{k}\)

Find a set of cylindrical coordinates of the point having the given cartesian coordinates: (a) \((4,4,-2)\) (b) \((-3 \sqrt{3}, 3,6)\) (c) \((1,1,1)\)

The given points \(A\) and \(B\) are opposite vertices of a rectangular parallelepiped, having its faces parallel to the coordinate planes. In each problem (a) draw a sketch of the figure, (b) find the coordinates of the other six vertices, (c) find the length of the diagonal \(A B\). \(A(-1,1,2) ; B(2,3,5)\)

Find an equation of the plane containing the given three points. $$ (3,4,1),(1,7,1),(-1,-2,5) $$

Draw a sketch of the given plane and find two unit vectors which are normal to the plane. $$ 4 x-4 y-2 z-9=0 $$

Find an equation of the surface of revolution generated by revolving the given plane curve about the indicated axis. Draw a sketch of the surface. \(x^{2}=4 y\) in the \(x y\) plane, about the \(x\) axis.

Find an equation of the surface of revolution generated by revolving the given plane curve about the indicated axis. Draw a sketch of the surface. \(x^{2}+4 z^{2}=16\) in the \(x z\) plane, about the \(x\) axis.

Find the values of \(k\) for which the intersection of the plane \(x+k y=1\) and the elliptic hyperboloid of two sheets \(y^{2}-x^{2}-z^{2}=1\) is (a) an ellipse and (b) a hyperbola.

Find the moving trihedral and the curvature at any point of the curve \(\mathbf{R}(t)=\cosh t \mathbf{i}+\sinh t \mathbf{j}+t \mathbf{k}\).

Find the vertices of the triangle whose sides have midpoints at \((3,2,3),(-1,1,5)\), and \((0,3,4)\).