Vectors in Two-Dimensional Space

  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. 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?
  3. 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 \(\bm{v}?\)
  4. Sketch the vectors \(\bm{u} = \langle 1,3 \rangle\) and \(\bm{v} = \langle -4,2 \rangle.\) Then sketch these vectors, and calculating them explicitly in terms of their components.
    \( \bm{u} + \bm{v} \)
    \( -\bm{u} \)
    \( \bm{u} - \bm{v} \)
    \( 4\bm{u} \)
    \( \bm{v} + 2\bm{u} \)
    \( 2\bm{u} - 3\bm{v} \)
    \( \bm{v} + \langle 7,-2 \rangle \)
    \( \bm{u} + 3\mathbf{j} \)
  5. What angle does \(\bm{u} = \langle 1,3 \rangle\) make with the positive \(x\)-axis? What angle does \(\bm{v} = \langle -4,2 \rangle\) make with the positive \(x\)-axis?
  6. What is the vector of magnitude \(15\) that makes an angle of \(63°\) clockwise from the positive \(x\)-axis?
  7. What is the vector of magnitude \(8\) that makes an angle of \(48°\) counterclockwise from the positive \(y\)-axis?
  8. If \(|\bm{u}| = 5\) and \(|\bm{v}| = 7\) what, if anything, can you conclude about \(|\bm{u} + \bm{v}|?\)