Midterm I Review Sheet for Math 207, Spring 2007

Concepts to know

1. The idea of a vector?
2. The geometric idea of vector addition, v+w
3. The geometric idea of scalar multiplication, kv
4. The geometric concept of the norm of a vector, ||v||
5. The geometric concept of the dot product of two vectors, v·w
6. The relationship of the dot product to the norm
7. The projection of one vector onto another
8. Geometric properties of the cross product of two vectors, v×w
9. Algebraic properties of (combinations of) the dot product, scalar product and cross product
10. The words orthogonal and normal
11. An equation of a plane
12. Equations of a line in space (parametric and vector forms)

Things to know how to do

1. compute components of a vector connecting two points in the plane or in space
2. computer components of v+w
3. computer components of kv
4. compute ||v|| from the components of v
5. compute the dot product v·w with two different approaches:
   1. using the components
   2. using the geometric approach, in terms of the lengths of v and w and the angle between them
6. compute the cross product v×w in two ways:
   1. using the complicated formula entirely with components
   2. using the geometric approach: it's direction (use the right hand rule!) and magnitude
7. tell if two vectors are parallel
8. tell if two vectors are orthogonal
9. tell if a vector is orthogonal to a plane
10. tell if a vector is parallel to a plane
11. find the angle between two vectors
12. find a vector along a line given the parametric or vector form of the line
13. find a normal vector to a plane given its equation (or given three points on it)
14. write down the equation of a plane given a normal and a point
15. find the area of a parallelogram
16. find the volume of a parallelepiped
17. find a vector parallel ("along", "in the direction of") a line, given an equation for the line
18. find a point on a line, given its equation
19. find a point on a plane, given its equation