Vectors in Two-Dim. Space

Exercises

  1. If you start at the initial point \((7,1)\) and travel to the terminal point \((2,5),\) along what vector did you travel?
  2. Let \(A\) be the point \((2,-9)\) and let \(B\) be the point \((3,4)\) What are the components of the vector \(\overrightarrow{AB}?\) What are the components of the vector \(\overrightarrow{BA}?\)
  3. If you start at the initial point \((-2,7)\) and travel along a vector \(\langle -1, 3 \rangle,\) what are the coordinates of the terminal point?
  4. Given points \(A\) and \(B\) such that the coordinates of \(A\) are \((-3,11)\) and such that \(\overrightarrow{AB} = \bigl\langle 2,-6 \bigr\rangle\) what must the coordinates of \(B\) be?
  5. Let \(\bm{v} = \langle 3,2 \rangle.\) What is the magnitude of \(\bm{v}?\) What is \(\bm{\hat{v}},\) the unit vector in the direction of \(\bm{v}?\)
  6. Sketch the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle\) in the \(xy\)-plane. Then sketch these vectors, and calculating them explicitly in terms of their components.
    \( \bm{u} + \bm{v} \)
    \( -\bm{u} \)
    \( \bm{u} - \bm{v} \)
    \( 4\bm{u} \)
    \( 2\bm{u} - 3\bm{v} \)
    \( \bm{v} + \langle 7,-2 \rangle \)
    \( \bm{u} + 3\mathbf{j} \)
  7. What angle does \(\bm{u} = \langle 1,3 \rangle\) make with the positive \(x\)-axis?
  8. What angle does \(\bm{v} = \langle -4,2 \rangle\) make with the positive \(y\)-axis?
  9. What is the vector of magnitude \(15\) that frames an angle of \(63°\) with the positive \(x\)-axis?
  10. What is the vector of magnitude \(8\) that frames an angle of \(148°\) with the positive \(x\)-axis?
  11. If \(|\bm{u}| = 5\) and \(|\bm{v}| = 7\) what, if anything, can you conclude about \(|\bm{u} + \bm{v}|?\)

Problems & Challenges

  1. Suppose an airplane plans to travel 456 mph with a heading of N30°W. How do you describe this velocity/trajectory as a vector? I.e. what are the east/west and north/south components of the plane’s velocity?
  2. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph. What is the plane’s heading as an angle? What is the plane’s speed?
  3. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but there is a steady 40 mph wind blowing from the southeast. What is the course of the plane as a vector? What is the actual speed of the plane?
  4. Suppose an airplane is flying at a constant speed of 430 mph and has an intended course due east-northeast. However there is a 40 mph wind blowing from the north. What should the plane’s heading be to stay on course?
  5. Suppose an airplane has a velocity vector of \(\langle 234, 389 \rangle\) mph, but due to the effects of a heavy wind, actually has a course of \(\langle 255, 377 \rangle\) mph. What is the velocity vector of the wind? What is the speed of the wind?
  6. Suppose you must swim across a straight river that is 20 ft wide, and you want to end up at the spot directly across the river from where you are now. The river has a constant current speed of 3 ft/s, but luckily you can swim at about 5 ft/s. In what direction do you need to swim to counteract the current to ensure you land directly across the river?
  7. Devise a formula for the distance from a point \(P\) to a line \({Ax\!+\!By\!=\!C.}\)