Plot the points \((1,4,-6)\) and \((-2,3,6)\) and \((3,-4,-5)\)
in \(xyz\)-space and determine which one is closest to the origin.
What’s the distance between the points
\((-6,7,-8)\) and \((0,21,-28)?\)
What are the coordinates of the midpoint between those points?
Sketch the planes
\(y=5\) and \(z=-3\) and \(x=y\)
in \(xyz\)-space.
What’s an equation of the sphere
with center \((4,5,-6)\) and radius three?
If you start at the initial point \((3,7,1)\)
and travel to the terminal point \((1,2,5),\)
along what vector did you travel?
If you start at the initial point \((-2,7,6)\)
and travel along a vector \(\langle -1,3,0 \rangle,\)
what are the coordinates of the terminal point?
Sketch the vectors
\(\bm{u} = \langle 3,-3,5 \rangle\)
and \(\bm{v} = \langle 2,5,-4 \rangle\)
in \(xyz\)-space,
then explicitly calculate each of the following:
\( \bm{u} + \bm{v} \)
\( \bm{u} - \bm{v} \)
\( |\bm{u}| \)
\( |\bm{v}| \)
\( \bm{\hat{u}} \)
\( \bm{\hat{v}} \)
\( 3\bm{u} \)
\( \bm{u}\cdot\bm{v} \)
\( \bm{u} - 2\bm{v} \)
\( \bm{v} + \langle 1,-2,3 \rangle \)
\( \bm{u} + 2\mathbf{j}-\mathbf{k} \)
What is the angle between the vectors \(\bm{u}\) and \(\bm{v}?\)
What are the three acute angles that \(\bm{u}\) makes with the coordinate axes?
What are the three acute angles that \(\bm{v}\) makes with the coordinate axes?