Section 11.3 : Dot Product

For problems 1 – 3 determine the dot product, \(\vec a\centerdot \vec b\).

1. \(\vec a = \left\langle {9,5, - 4,2} \right\rangle \), \(\vec b = \left\langle { - 3, - 2,7, - 1} \right\rangle \) Solution
2. \(\vec a = \left\langle {0,4, - 2} \right\rangle \) , \(\vec b = 2\vec i - \vec j + 7\vec k\) Solution
3. \(\left\| {\vec a} \right\| = 5\), \(\displaystyle \left\| {\vec b} \right\| = \frac{3}{7}\) and the angle between the two vectors is \(\displaystyle \theta = \frac{\pi }{{12}}\). Solution

For problems 4 & 5 determine the angle between the two vectors.

1. \(\vec v = \left\langle {1,2,3,4} \right\rangle \), \(\vec w = \left\langle {0, - 1,4, - 2} \right\rangle \) Solution
2. \(\vec a = \vec i + 3\vec j - 2\vec k\), \(\vec b = \left\langle { - 9,1, - 5} \right\rangle \) Solution

For problems 6 – 8 determine if the two vectors are parallel, orthogonal or neither.

1. \(\vec q = \left\langle {4, - 2,7} \right\rangle \), \(\vec p = - 3\vec i + \vec j + 2\vec k\) Solution
2. \(\vec a = \left\langle {3,10} \right\rangle \), \(\vec b = \left\langle {4, - 1} \right\rangle \) Solution
3. \(\vec w = \vec i + 4\vec j - 2\vec k\), \(\vec v = - 3\vec i - 12\vec j + 6\vec k\) Solution
4. Given \(\vec a = \left\langle { - 8,2} \right\rangle \) and \(\vec b = \left\langle { - 1, - 7} \right\rangle \) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec a}}\,\vec b\). Solution
5. Given \(\vec u = 7\vec i - \vec j + \vec k\) and \(\vec w = - 2\vec i + 5\vec j - 6\vec k\) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u\). Solution
6. Determine the direction cosines and direction angles for \(\displaystyle \vec r = \left\langle { - 3, - \frac{1}{4},1} \right\rangle \). Solution