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### Section 5-3 : 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 $$\vec v\centerdot \vec v = {\left\| {\vec v} \right\|^2}$$.