Task 2.1

Start by looking up the terms vector-valued function and vector parameterization of a curve.

  1. Write definitions in your own words for the terms above.
  2. For each vector parameterization below, construct a graph of the curve. [Hint: make a table of points if needed, including $t$, $x$, and $y$, and then just plot the $(x,y)$ coordinates).
    1. $\vec r(t) = \left< 2t+1, 4-3t\right>$ for $0\leq t\leq 2$.
    2. $(x,y) = (\cos t, \sin t)$ for $0\leq t\leq 3\pi/2$.
    3. $(x,y)(t) = (\sin t, \cos t)$ for $0\leq t\leq \pi$.
    4. $\langle x,y,z\rangle = (2\cos t, 2\sin t, t)$ for $0\leq t\leq 4\pi$.

Task 2.2

Start by locating a definition of the dot product of two vectors and what it means for two vectors to be orthogonal, as well as the dot product form of the law of cosines.

  1. Compute the dot product of the two vectors $\vec a = 3\vec i-4\vec j+2\vec k$ and $\vec b = (-1,3,6)$.
  2. Find the angle between $\vec a$ and $\vec b$.
  3. Give a value $k$ so that the vectors $\vec a = 3\vec i-4\vec j+2\vec k$ and $\vec c = \langle 2, -1, k\rangle$ are orthogonal.
  4. A car is moving in the direction $\vec v = (-5,7)$. The car makes a 90 degree turn to the left. Give a vector that is parallel to this new direction of motion.

Task 2.3

Start by looking up the definition of a unit vector. Consider the two points $P = (1, 2, 3)$ and $Q = (2, −1, 0)$.

  1. Write the vector $\vec {PQ} $ in component form $(a, b, c)$.
  2. Find the length of vector $\vec {PQ} $.
  3. Find a unit vector in the same direction as $\vec {PQ} $.
  4. Find a vector of length 7 units that points in the same direction as $\vec {PQ} $.

Task 2.4

The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below.

  1. In section 1.1, complete checkpoints 1.1, 1.2, and 1.3. Use the corresponding examples, if needed, to help you.
  2. In section 2.3, complete an exercise for each group in 123-144, and then try a few problems in 149-154.