Vectors in Three-Dim. Space

Exercises

  1. If you start at the initial point \((3,7,1)\) and travel to the terminal point \((1,2,5),\) along what vector did you travel?
  2. Let \(A\) be the point \((2,-9,1)\) and let \(B\) be the point \((3,2,-4)\) 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 \((-2,7,6)\) and travel along a vector \(\langle -1,3,0 \rangle,\) what are the coordinates of the terminal point?
  4. Given points \(A\) and \(B\) such that the coordinates of \(A\) are \((0,-3,1)\) and such that \(\overrightarrow{AB} = \bigl\langle 10,2,-5 \bigr\rangle\) what must the coordinates of \(B\) be?
  5. Let \(\bm{v} = \langle 3,2,-4 \rangle.\) What is the magnitude of \(\bm{v}?\) What is \(\bm{\hat{v}},\) the unit vector in the direction of \(\bm{v}?\)
  6. What angle does \(\bm{v} = \langle 3,0,2 \rangle\) make with the \(z\)-axis?
  7. What angle does \(\bm{u} = \langle 1,3,-2 \rangle\) make with the \(xy\)-plane?
  8. Sketch the vectors \(\bm{u} = \langle 3,-3,5 \rangle\) and \(\bm{v} = \langle 2,5,-4 \rangle\) in \(xyz\)-space, then explicitly calculate each of the following:
    \( \bm{u} + \bm{v} \)
    \( \bm{u} - \bm{v} \)
    \( |\bm{u}| \)
    \( |\bm{v}| \)
    \( \bm{\hat{u}} \)
    \( \bm{\hat{v}} \)
    \( 3\bm{u} \)
    \( \bm{u}\cdot\bm{v} \)
    \( \bm{u} - 2\bm{v} \)
    \( \bm{v} + \langle 1,-2,3 \rangle \)
    \( \bm{u} + 2\mathbf{j}-\mathbf{k} \)
    What are the three acute angles that \(\bm{u}\) makes with the coordinate axes? What are the three acute angles that \(\bm{v}\) makes with the coordinate axes? What is the measure of the angle between the vectors \(\bm{u}\) and \(\bm{v}?\) What is the area of the triangle framed by \(\bm{u}\) and \(\bm{v}?\)
  9. What is the measure of angle between \(\bm{u} = \langle 7,-2,1\rangle\) and \(\bm{v} = \langle 3,5,-3 \rangle?\) What is the area of the triangle framed by those vectors?
  10. What is the area of the triangle in space with vertices located at the coordinates \((1,1,1)\) and \((3,-3,8)\) and \((2,7,0)?\)