# Recitation 1

Section 10.1, Exercise 15: Show that the equation $2x^2+2y^2+2z^2=8x-24z+1$ represents a sphere, and find its center and radius.

Comment: The idea is to complete the squares. Once the equation is written in the canonical form $(x-a)^2+(y-b)^2+(z-c)^2=r^2$, the center is just $(a,b,c)$ and the radius is $r$.

Section 10.1, Exercise 37: Find the distance between the spheres $x^2+y^2+z^2=4$ and $x^2+y^2+z^2=4x+4y+4z-11$

Comment: If two spheres whose centers are $O_1, O_2$ and radii $r_1, r_2$ are apart, the distance between them is given by $|O_1O_2|-(r_1+r_2)$. Note that this formula doesn’t work if the spheres intersect or one is contained in the other.

Section 10.2, Exercise 2: Write each combination of vectors as a single vector. $\vec{AB}+\vec{BC}$, $\vec{CD}+\vec{DB}$, $\vec{DB}-\vec{AB}$, $\vec{DC}+\vec{CA}+\vec{AB}$.

Solution: $\vec{AB}+\vec{BC}=\vec{AC}$, $\vec{CD}+\vec{DB}=\vec{CB}$, $\vec{DB}-\vec{AB}=\vec{DA}$, $\vec{DC}+\vec{CA}+\vec{AB}=\vec{DB}$.

Section 10.2, Exercise 18: Find a vector that has the same direction as $\langle 2, 4, 2\rangle$ but has length $6$.

Comment: The quickest way to find the vector is to find the scale factor. The length of the original vector is $\sqrt{24}$. Thus the scale factor is $6/\sqrt{24}$. Multiplying the scale factor to the vector $(2,4,2)$ gives the answer.

Section 10.2, Exercise 25: Find the magnitude of the resultant force and the angle it makes with the positive x-axis. (see p.550 for picture)

Solution: Let’s call the force pointing to the left $F$ and the other one $G$. With some effort, $F=(-300, 0)$ and $G=(100,100\sqrt{3})$. Now $F+G=(-200, 100\sqrt{3})$. The magnitude is $\sqrt{200^2+100^2\times 3}=100\sqrt{7}$ and the angle it makes with the positive x-axis is $2\pi - \arctan \sqrt{3}/2$.