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Section 11.2 : 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 : \(\left( {a + b} \right)\vec v = a\vec v + b\vec v\).