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Section 5-2 : Vector Arithmetic

1. Given $$\vec a = 3\vec i - 9\vec j$$ and $$\vec b = - 6\vec i + \vec j$$ compute each of the following.
1. $$10\vec b$$
2. $$14\vec a + 20\vec b$$
3. $$\left\| {8\vec b - \frac{1}{3}\vec a} \right\|$$
2. Given $$\vec u = \left\langle {0,4, - 1} \right\rangle$$ and $$\vec v = \left\langle {6, - 2, - 7} \right\rangle$$ compute each of the following.
1. $$\frac{3}{4}\vec u$$
2. $$- 3\vec u - 7\vec v$$
3. $$\left\| {\vec v + 10\vec u} \right\|$$
3. Given $$\vec p = \left\langle {3, - 1, - 2} \right\rangle$$ and $$\vec q = - \frac{1}{3}\vec i - \frac{1}{2}\vec k$$ compute each of the following.
1. $$2\vec p$$
2. $$9\vec q - 2\vec p$$
3. $$\left\| {8\vec p - 12\vec q} \right\|$$
4. Find a unit vector that points in the same direction as $$\vec a = \left\langle {10, - 3,8, - 2} \right\rangle$$.
5. Find a unit vector that points in the same direction as $$\vec w = - \vec i - 6\vec j$$.
6. Find a unit vector that points in the opposite direction as $$\vec c = 2\vec i + 7\vec j - 5\vec k$$.
7. Find a unit vector that points in the opposite direction as $$\vec b = \left\langle {0, - 3, - 11} \right\rangle$$.
8. Find a vector that points in the same direction as $$\vec p = 2\vec i - 3\vec j + \vec k$$ with a magnitude of $$\frac{1}{2}$$.
9. Find a vector that points in the opposite direction as $$\vec a = \left\langle { - 3, - 14,2} \right\rangle$$ with a magnitude of 32.
10. Find a vector that points in the same direction as $$\vec b = - 3\vec i + 2\vec k$$ with a magnitude that is $$\frac{1}{{10}}$$ the magnitude of $$\vec b$$.
11. Determine if $$\vec p = 8\vec i - 3\vec j$$ and $$\vec q = 16\vec i - 6\vec j$$ are parallel vectors.
12. Determine if $$\vec v = \left\langle {1,0, - 4} \right\rangle$$ and $$\vec w = \left\langle {9,3,1} \right\rangle$$ are parallel vectors.
13. Determine if $$\vec a = 10\vec i + 8\vec j + 20\vec k$$ and $$b = \left\langle { - 35, - 28,70} \right\rangle$$ are parallel vectors.
14. Prove the property : $$\vec u + \left( {\vec v + \vec w} \right) = \left( {\vec u + \vec v} \right) + \vec w$$.
15. Prove the property : $$\vec v + \vec 0 = \vec v$$.
16. Prove the property : $$1\vec v = \vec v$$.
17. Prove the property : $$\left( {a + b} \right)\vec v = a\vec v + b\vec v$$.