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Section 11.3 : Dot Product

For problems 1 – 5 determine the dot product, \(\vec a\centerdot \vec b\).

1. \(\vec a = 9\vec i - 8\vec k\), \(\vec b = \left\langle {3, - 2,1} \right\rangle \)
2. \(\vec a = \left\langle {4, - 1,0,5} \right\rangle \) , \(\vec b = \left\langle {3,0, - 10,6} \right\rangle \)
3. \(\vec a = \vec i - 5\vec j - 2\vec k\) , \(\vec b = - 4\vec i + 2\vec j + 8\vec k\)
4. \(\displaystyle \left\| {\vec a} \right\| = \frac{1}{4}\), \(\displaystyle \left\| {\vec b} \right\| = \frac{9}{4}\) and the angle between the two vectors is \(\theta = \pi \).
5. \(\displaystyle \left\| {\vec a} \right\| = 24\), \(\left\| {\vec b} \right\| = 9\) and the angle between the two vectors is \(\displaystyle \theta = \frac{{2\pi }}{7}\).

For problems 6 – 8 determine the angle between the two vectors.

1. \(\vec p = 9\vec i - \vec j\), \(\vec q = - 3\vec i - 6\vec j\)
2. \(\vec a = \left\langle {4,0, - 3} \right\rangle \), \(\vec b = 2\vec i + 10\vec j - 11\vec k\)
3. \(\vec w = \left\langle {8,3, - 1, - 4} \right\rangle \), \(\vec v = \left\langle { - 1,9,4, - 8} \right\rangle \)

For problems 9 – 12 determine if the two vectors are parallel, orthogonal or neither.

1. \(\vec q = 7\vec i - 14\vec j - 21\vec k\), \(\vec p = \left\langle { - 4,8,12} \right\rangle \)
2. \(\vec u = \left\langle {5,0, - 2} \right\rangle \), \(\vec q = \left\langle {4, - 7,10} \right\rangle \)
3. \(\vec a = 9\vec i - \vec j + 5\vec k\), \(\vec b = - 2\vec i + 7\vec j + \vec k\)
4. \(\vec v = \left\langle { - 1,3,1,5} \right\rangle \), \(\vec w = \left\langle { - 8,3, - 7, - 2} \right\rangle \)
5. Given that \(\vec a\centerdot \vec b = - 6\), \(\left\| {\vec a} \right\| = 4.3\) and the angle between \(\vec a\) and \(\vec b\) is \(\displaystyle \theta = \frac{\pi }{6}\) determine if \(\vec b\) is a unit vector or not.

For problems 14 & 15 determine the value of b for which the two vectors will be orthogonal.

1. \(\vec u = \left\langle {3, - 1,6} \right\rangle \), \(\vec v = \left\langle {3, - 2b,1} \right\rangle \)
2. \(\vec u = \left\langle {1 - b,4, - 2} \right\rangle \), \(\vec v = \left\langle {b,6,3b} \right\rangle \)
3. Given \(\vec a = \vec i + 3\vec j - 2\vec k\) and \(\vec b = - 3\vec i - 4\vec j + 7\vec k\) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec a}}\,\vec b\).
4. Given \(\vec a = \vec i + 3\vec j - 2\vec k\) and \(\vec b = - 3\vec i - 4\vec j + 7\vec k\) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec b}}\,\vec a\).
5. Given \(\vec p = \left\langle {5, - 2,1} \right\rangle \) and \(\vec q = \left\langle {0,4,8} \right\rangle \) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec p}}\,\vec q\).
6. Given \(\vec u = \left\langle {1,3,0, - 2} \right\rangle \) and \(\vec w = \left\langle { - 2,2,4,1} \right\rangle \) compute \({{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u\).
7. Determine the direction cosines and direction angles for \(\vec r = \left\langle {5,2, - 7} \right\rangle \).
8. Determine the direction cosines and direction angles for \(\displaystyle \vec r = \left\langle {\frac{1}{2}, - \frac{3}{4},\frac{5}{2}} \right\rangle \).
9. Prove the property \(\left( {c\vec v} \right)\centerdot \vec w = \vec v\centerdot \left( {c\vec w} \right)\).
10. Prove the property \(\vec v\centerdot \vec w = \vec w\centerdot \vec v\).
11. Prove the property \(\vec v\centerdot \vec 0 = 0\).
12. Prove the property \(\vec v\centerdot \vec v = {\left\| {\vec v} \right\|^2}\).