In-class work for 5 June 2012

1. What restrictions must be made on $x$, $y$, and $z$ so that the triple $(x,y,z)$ will represent a point on...

    a.) the $y$-axis?

    b.) the $xz$-plane?

2. Find the distance from the point $(2,6,1)$ to the point...

    a.) $(3,1,2)$.

    b.) $(-5,2,-2)$.

3. Compute the dot product of vectors...

    a.) $< 1,2,3 >$ and $< 4,5,6 >$.

    b.) $< -5,8,-4 >$ and $< 7,8,9 >$.

4. Compute the cross product of vectors...

    a.) $<1,2,3>$ and $<4,5,6>$.

    b.) $<1,-2,1>$ and $<2,1,1>$.

5. Let $\vec{x} = < x_1,x_2,x_3 >$ where $x_1, x_2,$ and $x_3$ are real numbers.

    a.) What is $\vec{x} \cdot \vec{x}$?

    b.) How does your answer to part a.) relate to the magnitude of the vector $\vec{x}$?

    c.) Why?

6. Find the angle between the vectors $<0,1>$ and $<1,1>$. (no calculator necessary!)

7. What would be the projection of the point $(3,1,-4)$ onto...

        a.) the $xy$-plane?

        b.) the $yz$-plane?

8. Find the center and radius of the circle given by $$x^2-2x+y^2-4y+z^2-6z=-13.$$