Problem 1
Find the center and the radius of the sphere: $x^{2}+y^{2}+z^{2}+4x-4z=0$. Sketch the graph of the sphere.
Solution:
We know that in general sphere has the form $(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}$ where $(x_{0},y_{0},z_{0})$ is the center of the sphere
and $r$ is the radius of the sphere. So let's try to obtain such form in our question by simple algebraic manipulations.
$x^{2}+y^{2}+z^{2}+4x-4z = (x^{2}+4x+4)-4 + y^{2} + (z^{2} - 4z+4)-4 = (x+2)^{2}+y^{2}+(z-2)^{2}-8$
On the other hand it must be zero. So let's take $-8$ to another side. Thus we obtain: $(x+2)^{2}+y^{2}+(z-2)^{2}=8=(\sqrt{8})^{2}$
Answer: Center of the spehre is $(-2,0,2)$ and the radius of the sphere is $\sqrt{8}=2\sqrt{2}$.
You also need to draw a picture. In order to draw a picture you need to indicate the center of the sphere in 3 dimensional space and draw a sphere around this point.
Problem 2
Let $A=(-1,2,0)$ and $B=(3,1,-2)$. Express the vector $\vec{AB}$ in its standart form: $\vec{v} = v_{1}\vec{i}+v_{2}\vec{j}+v_{3}\vec{k}$.
Solution:
$\vec{AB} = \vec{AO}+\vec{OB} = -\vec{OA}+\vec{OB}=-\langle-1,2,0 \rangle + \langle3,1,-2 \rangle=\langle 4,-1,-2\rangle =4\vec{i}-\vec{j}-2\vec{k}$
Problem 3
Let $\vec{u}=\langle 5,12,0\rangle$ and $\vec{v}=\langle \frac{3}{5},0,\frac{4}{5}\rangle$.
Find the dot product $\vec{u}\cdot \vec{v}$ and find the vector projection of $\vec{u}$ onto $\vec{v}$ i.e. $\text{Proj}_{\vec{v}}\vec{u}$.
Solution:
$$
\vec{u}\cdot \vec{v} = 5 \cdot \frac{3}{5}+12\cdot 0 +0\cdot \frac{4}{5}=3
$$
$$
\text{Proj}_{\vec{v}}\vec{u} = \frac{\vec{u}\cdot \vec{v}}{|\vec{v}|^{2}}\vec{v}=\frac{3}{\sqrt{\left(\frac{3}{5}\right)^{2}+0^{2}+\left(\frac{4}{5}\right)^{2}}} \left \langle\frac{3}{5},0,\frac{4}{5} \right\rangle=\left\langle\frac{9}{5},0,\frac{12}{5} \right\rangle
$$