91Ó°ÊÓ

Prove that for all vectors \(\boldsymbol{u}\) and \(\boldsymbol{v},(\boldsymbol{u} \times \boldsymbol{v}) \cdot \boldsymbol{v}=0 .\)

Find an equation for the sphere with radius 1 and center at (0,1,0) in spherical coordinates.

Find an equation of the sphere with center at (2,-1,3) and radius \(5 .\)

Find an equation of the sphere with center (3,-2,1) and that goes through the point (4,2,5)

Suppose the curve \(z=x\) in the xz-plane is rotated around the z-axis. Find an equation for the resulting surface in spherical coordinates.

Find \(|\boldsymbol{v}|, \boldsymbol{v}+\boldsymbol{w}, \boldsymbol{v}-\boldsymbol{w},|\boldsymbol{v}+\boldsymbol{w}|,|\boldsymbol{v}-\boldsymbol{w}|\) and \(-2 \boldsymbol{v}\) for \(\boldsymbol{v}=\langle 3,2,1\rangle\) and \(\boldsymbol{w}=\langle-1,-1,-1\rangle .\)

Find the cosine of the angle between \langle 2,0,0\rangle and \langle-1,1,-1\rangle ; use a calculator if necessary to find the angle.

Define the triple product of three vectors, \(\boldsymbol{x}, \boldsymbol{y},\) and \(z,\) to be the scalar \(\boldsymbol{x} \cdot(\boldsymbol{y} \times \boldsymbol{z}) .\) Show that three vectors lie in the same plane if and only if their triple product is zero. Verify that \(\langle 1,5,-2\rangle,\) \langle 4,3,0\rangle and \langle 6,13,-4\rangle all lie in the same plane.

Let \(P=(4,5,6), Q=(1,2,-5) .\) Find \(\overrightarrow{P Q}\). Find a vector with the same direction as \(\overrightarrow{P Q}\) but with length 1 . Find a vector with the same direction as \(\overrightarrow{P Q}\) but with length \(4 .\)

Find the angle between the diagonal of a cube and one of the edges adjacent to the diagonal.