1.

Let ${\bf v}=<1,\sqrt{3}>, {\bf u}= <3,2>.$Find ${\bf v}\cdot {\bf u}$
Find the angle between ${\bf v}$ and x axis.
Find the angle between ${\bf v}$ and y axis.
Find the angle between ${\bf v}$ and ${\bf u}$. You may leave the answer in terms of $\cos ^{-1}$.
Draw ${\bf v}, {\bf u}, {\bf v}+ {\bf u}, 2 {\bf v}, 0.2 {\bf u}$.

2.

Find the following if it is defined. ${\bf u}=<1,-2,1>, {\bf v}= <1,2,3>, {\bf w}= <-1,1,2>$, $~{\bf i}=<1,0,0>, ~{\bf j}=<0,1,0>,~{\bf k}=<0,0,1>$.

(a)

${\bf u}+ \v0$.

(b)

${\bf u}+ {\bf v}$.

(c)

$10 {\bf v}$.

(d)

${\bf u}\cdot {\bf v}$.

(e)

${\bf v}\cdot {\bf w}\cdot {\bf u}$.

(f)

${\bf u}\times {\bf v}$.

(g)

${\bf u}\times {\bf v}\times ~{\bf i}$.

(h)

$~{\bf i}\times ~{\bf i}$.

(i)

The angle between ${\bf v}$ and ${\bf w}$.

(j)

$\Vert {\bf v}\Vert$.

(k)

Find the unit vector parallel to ${\bf v}$.

3.

With the same vectors as above.

(a)

Find the orthogonal projection of ${\bf v}$ on $~{\bf i}$.

(b)

Find the orthogonal projection of ${\bf v}$ on ${\bf u}$.

(c)

Find the vector component of ${\bf v}$ orthogonal to ${\bf u}$.

(d)

The angle beween the projection of ${\bf v}$ onto xy plane and x axis.

(e)

The angle between z axis and vv.

(f)

The plane passing (10,20,30) and perpendicular to ${\bf v}$.

(g)

Any vector perpendicular to ${\bf v}$ and ${\bf w}$.

(h)

The plane passing through ${\bf v}$ and ${\bf w}$.

4.

Find the vector $\overline{P_1 P_2}$, where P₁(1,2,3), P₂(2,3,4).

5.

Find the equation of the circle in xy plane whose center is (0,0) and radius is 2.
Find the tangent line to this circle at $(\sqrt{2}, \sqrt{2})$.

6.

Find the equation of the sphere in 3 space, whose center is (1,2,3) and radius is 2.
Find the tangent plane to this sphere at (3,2,3).
Find the intersection of this sphere and z=0.

7.

Find the equation of the cylinder whose base lie on yz plane and base has the radius 2 and passes (0, 2,0).
Find the intersection of this cylinder with yz plane.

8.

Describe the surface defined by

(a)

x² + y² + z² + 10x + 4y + 2z -10=0.

(b)

z² + (y-2)² = 9.

(c)

Determine if the line

x= 3 + 8 t, y= 4 + 5 t , z= -3 - t

is parallel to the plane x - 3y + 5z =12.

9.

Coordiante sysytem.

(a)

You may use a calculator. Let X=(x,y,z) = (1,3,4). Find the angle between X and z axis.
Indicate this angle in the figure below.
Find the projection of X onto xy plane. Call this point X₁2
Plot this point in the figure below.
Find the angle between x axis and X₁2.
Plot this angle.
Find the cylidraical coordinate of X.
Find the spherical coordinate of X.

10.

In spherical cooridinate a surface has the equation, $\rho = 2$. What is this surface?

11.

Write the cylindrical coordinate of

$$z = \sqrt{3 x^2 + 3 y^2}$$

12.

All Quizzes.

1.

Find the equation of the line passing through P₁(1,2,3) and P₂(2, 2.5, 3.3).

2.

Find the derivative.

(a)

A circular helix in 3 space given by $\r (t) = ( 2 \cos t, 2 \sin t , 4 t)$ at t=$\pi$.

(b)

$\r (t) = ( 3, 5, 7)$ at $t= \pi$.

(c)

$\r (t) = ( t^3, t^2, t)$ at t = 2.

3.

Integrate.

(a)

$\int ( \sin t ~{\bf i}+ \cos t ~{\bf k}) d t$.

(b)

$\int_0^2 (\sin t, \cos t , t^2) dt$

(c)

$\int_1^x (t, t^2, t^3) dt$.

4.

Consider a circular helix in 3 space given by $\r (t) = ( \cos t, \sin t , 4 t)$ and $~{\bf s}(t) = ( t^2, t, 3 t)$. Find the following.

(a)

$\frac{d}{dt} 3 \r (t)$.

(b)

$\frac{d}{d t}\r (t) + ~{\bf s}(t)$.

(c)

$\frac{d}{d t}\cos t ~{\bf r}(t)$.

(d)

$\int_0^2 (~{\bf r}(t) - ~{\bf s}(t)) d t$

(e)

$\int_0^3 (10 ~{\bf r}(t)) dt$.

5.

Let $~{\bf r}(t) = (1, t, 2 t)$. Find $~{\bf r}(0), ~{\bf r}(0.5), ~{\bf r}(0.05), ~{\bf r}(0.01)$.
Find the equation of the line passing through $~{\bf r}(0)$ and

(a)

$~{\bf r}(0.5)$

(b)

$~{\bf r}(0.05)$

(c)

$~{\bf r}(0.001)$

(d)

Write any mathematical thought about this problem.

6.

Let $\r (t) = ( \cos t, \sin t , 4 t)$ and $~{\bf s}(t) = ( \cos t, \sin t, 2)$. Find $\r (0)$.
Find $\frac{d}{d t}\r (t)$ at t=0.
Find the equation of the line passes through $\r (0)$ and parallel to $\r ' (0)$.
Find the tangent line to $\r (t)$ at t=0.
Find the tangent line to $~{\bf s}(t)$ at t=0.