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### Section 5-3 : 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