Exercises


  1. Given that \( \vec{u} = s\hat{i} + t\hat{j} + d\hat{k} \) and \( \vec{v} = p\hat{i} + q\hat{j} + r\hat{k} \), show how
    1. the vector \( \vec{u} - \vec{v} \) can be written in the form \( a\hat{i} + b\hat{j} + c\hat{k} \).
    2. the vector \( m\vec{u} \) can be written in the form \( a\hat{i} + b\hat{j} + c\hat{k} \).
  2. Given that \( \vec{u} = 3\hat{i} - \hat{j} + 4\hat{k} \) and \( \vec{v} = -\hat{i} + 2\hat{j} + 5\hat{k} \), write each of the following in the form \( a\hat{i} + b\hat{j} + c\hat{k} \).
    1. \( 5 \vec{u} \)
    2. \( \vec{u} + \vec{v} \)
    3. \( -2\vec{u} + \vec{v} \)
    4. \( 3(\vec{u} - 2\vec{v}) \)
  3. In each case, determine vector \( \overrightarrow{AB} \).
    1. \( A~(2, -1), B~(2, -5) \)
    2. \( A~(1, 7), B~(2, -4) \)
    3. \( A~(3, 5, 1), B~(-6, 5, -2) \)
    4. \( A~(4, -2, 4), B~(8, -10, 6) \)
  4. Determine if the following set of points is collinear.
    1. \( P~(2, -1), Q~(8, 6), R~(14, 10) \)
    2. \( E~(-4, 3), F~(-2, 2), G~(-6, 4) \)
    3. \( M~(5, 1, -2), N~(8, 7, 4), P~(14, 19, 16) \)
    4. \( X~(2, -1, 4), Y~(3, 4, -2), Z~(1, -1, 10) \)
  5. Prove that \( A~(-5, 9, 3), B~(5, 7, 9) \), and \( C~(20, 4, 18) \) are collinear. What is the ratio \( \left \lvert \overrightarrow{AB} \right \rvert : \left \lvert \overrightarrow{BC} \right \rvert \)?
  6. \( A~(2, -1, 4), B~(6, 2, 8) \), and \( C~(3, -2, 5) \) are three of the vertices of parallelogram \( ABCD \). Determine the coordinates of \( D \).
  7. Find a point on the \( x \)-axis which is equidistant from the points \( A~(1, -6, 1) \) and \( B~(4, 1, -3) \).
  8. A parallelepiped is a \( 3 \)-dimensional figure having \(6\) sides, each of which is a parallelogram. Given the points \( O~(0, 0, 0), X~(0, 3, -1), Y~(2, 0, 4) \), and \( Z~(3, -1, 0) \), consider a parallelepiped such that the vectors \( \overrightarrow{OX}, \overrightarrow{OY} \), and \( \overrightarrow{OZ} \) form three edges which meet at \( O \). Determine the coordinates of the other \( 4 \) vertices.