Exercises


  1. For each point \( Q \) given, write the position vector \( \overrightarrow{OQ} \) in terms of \( \hat{i} \) and \( \hat{j} \).
    1. \( Q~(3, -4) \)
    2. \( Q~(-5, -1) \)
  2. For each point \( Q \) in question 1, find the magnitude of the position vector \( \overrightarrow{OQ} \) and its direction relative to the positive \( x \)-axis.
  3. For each point \( R \) given, find the magnitude of the position vector \( \overrightarrow{OR} \).
    1. \( R~(4, -3, 12) \)
    2. \( R~(2, -1, 3) \)
  4. Write the position vectors of the point \( A \) shown, in the form \( a\hat{i} + b\hat{j} + c\hat{k} \).

    1. Rectangular prism completing the vector (-4, 3, 2)

    2. Rectangular prism completing the vector (5, -4, 3)
  5. Draw a sketch to show the point \( D~(4, 2, -3) \) and draw the position vector \( \overrightarrow{OD} \).
  6. Determine the direction angles for each of the following vectors.
    1. \( \vec{v} = 2\hat{i} - \hat{j} + 3\hat{k} \)
    2. \( \overrightarrow{OA} = (-1, 4, -5) \)
    3. \( \vec{u} = 5\hat{i} - 12\hat{k} \)
    4. \( \overrightarrow{OB} = (0, 3, -4) \)
  7. Each of the following general points lie on a coordinate axis and/or on a coordinate plane. If the point lies on a coordinate axis, state the axis on which it lies. For all other points, state the coordinate plane on which it lies.
    1. \( A~(x, y, 0) \)
    2. \( B~(0, 0, z) \)
    3. \( C~(x, 0, z) \)
    4. \( D~(0, y, 0) \)
    5. \( E~(0, y, z) \)
    6. \( F~(x, 0, 0) \)
  8. Given that \( \left \lvert \vec{v} \right \rvert = 3\sqrt{5} \) and the direction angles of \( \vec{v} \) are \( \alpha = 41.8^\circ, \beta = 107.3^\circ \), and \( \gamma = 126.6^\circ \), write \( \vec{v} \) in the form \( a\hat{i} + b\hat{j} + c\hat{k} \). Round the components of \( \vec{v} \) to the nearest integer.
  9. Find a unit vector parallel to each of the given vectors.
    1. \( \vec{v} = (2, -5) \)
    2. \( \overrightarrow{OZ} = \hat{i} - 2\hat{j} + 4\hat{k} \)
    3. \( \vec{w} = (-5, 12) \)
    4. \( \overrightarrow{OP} = 3\hat{i} + 3\hat{j} - \hat{k} \)
  10. Find the \( 3 \) direction angles for a vector whose direction angles are all equal.