What you must know for the test

Note. The Exam will consist of a problem from each of the following parts (five problems in total). Each problem will be worth 10 points.

Part I. Vectors:

1. Lenght of a vector $\vec v=a\vec\imath +b\vec\jmath +c\vec k=$ ?

2. Unit vector in the direction of the vector $\vec v=a\vec\imath +b\vec\jmath +c\vec k=$ ?
3. Given two points $P(x_0,y_0,z_0)$ and $Q(x_1,y_1,z_1)$, write the coordinates of the vector $\overset{\longrightarrow}{PQ}$.
4. Dot product of two vectors $\vec v=a_1\vec\imath +b_1\vec\jmath +c_1\vec k$, $\vec w=a_2\vec\imath +b_2\vec\jmath +c_2\vec k$:
   1. What is the dot product $\vec v\cdot{\vec w}=$ ?
   2. What does the dot product of two orthogonal vectors equal to?
   3. Is the dot product a scalar (scalar is another word for ``number'') or is it a vector?

5. Vector and scalar projection of a vector $\vec v$ onto a vector $\vec w$.

Part II. Cross product:

1. Cross product of two vectors $\vec v=a_1\vec\imath +b_1\vec\jmath +c_1\vec k$, $\vec w=a_2\vec\imath +b_2\vec\jmath +c_2\vec k$, write $\vec v\times\vec w$ as a determinant and then evaluate the determinant.

2. Is $\vec v\times\vec w$ a scalar or is it a vector?

3. What is the area of the parallelogram determined by two vectors $\vec v$ and $\vec w$?

4. Given $\vec v$ and $\vec w$, how can you find a vector which is orthogonal to both of them?

5. What is the area of a triangle with vertices $P$, $Q$ and $R$ in terms of the cross product of two vectors?

Part III. Planes and Lines:

1. Planes:
   1. Equation of the plane through the point $P(x_0,y_0,z_0)$ with normal vector $\vec N= A\vec\imath +B\vec\jmath +C\vec k$. Warning: you may not be given $P$ or $\vec N$ directly, but may have to find them from the data in the problem.
   2. Equation of a plane through three points $P$, $Q$ and $R$ which are not aligned (they form a triangle). This is similar to (5) in Part II in the sense that you have to find the cross product of two vectors in order to find a normal vector to the plane.
   3. Given an equation of a plane, sketch it.
2. Lines:
   1. Write the parametric equation of the line through the point $P(x_0,y_0,z_0)$ aligned to the vector $\vec v= a\vec\imath +b\vec\jmath +c\vec k$.
   2. If $a,b,c\neq 0$, write the parametric equation of the line through the point $P(x_0,y_0,z_0)$ aligned to the vector $\vec v= a\vec\imath +b\vec\jmath +c\vec k$.
   3. Find the points of intersection of a line with the coordinate planes.

Part IV. Quadric Surfaces:

1. Given the equation of a sphere written as $x^2+y^2+z^2+ax+by+cz+d=0$, complete squares to write it in the form $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=r^2$ and then identify the coordinates of the center and the radius.

2. Identify the equation and sketch the following:
   1. Ellipsoid.
   2. Elliptic paraboloid.
   3. Parabolic cylinder.

Part V. Distance:

1. Between two points.
2. Between a point and a plane.
3. Between a point and a line.