If $$ a=6 i+3 j $$ and $$ b=i-3 j $$, use the information to answer questions 12 and 13 . 12. Find $$ \mathrm{a} \cdot \mathrm{b} $$ ? $$ \begin{array}{llll} ext { (A) } 6 & ext { (B) } 3 & ext { (C) }-3 & ext { (D) }-6\end{array} $$ 13. Given $$ heta $$ that is the angle between a and $$ b $$, find $$ \cos heta $$ ? (A) $$ \begin{array}{llll}\frac{\sqrt{5}}{2} & ext { (B) }-\frac{1}{\sqrt{5}} & ext { (C) } \frac{\sqrt{2}}{5} & ext { (D) }-\frac{\sqrt{2}}{10}\end{array} $$

Question

If $$ a=6 i+3 j $$ and $$ b=i-3 j $$, use the information to answer questions 12 and 13 . 12. Find $$ \mathrm{a} \cdot \mathrm{b} $$ ? $$ \begin{array}{llll}	ext { (A) } 6 & 	ext { (B) } 3 & 	ext { (C) }-3 & 	ext { (D) }-6\end{array} $$ 13. Given $$ 	heta $$ that is the angle between a and $$ b $$, find $$ \cos 	heta $$ ? (A) $$ \begin{array}{llll}\frac{\sqrt{5}}{2} & 	ext { (B) }-\frac{1}{\sqrt{5}} & 	ext { (C) } \frac{\sqrt{2}}{5} & 	ext { (D) }-\frac{\sqrt{2}}{10}\end{array} $$

UpStudy AI · Accepted Answer

Let's solve each part step by step.

### Given:
- Vector **a** = $$6\mathbf{i} + 3\mathbf{j}$$
- Vector **b** = $$\mathbf{i} - 3\mathbf{j}$$

### 12. Find $$ \mathbf{a} \cdot \mathbf{b} $$
The dot product of two vectors $$\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}$$ and $$\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}$$ is calculated as:
$$
\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2
$$
Substituting the given values:
$$
\mathbf{a} \cdot \mathbf{b} = (6)(1) + (3)(-3) = 6 - 9 = -3
$$
**Answer:** (C) $$-3$$

### 13. Find $$ \cos \theta $$, the angle between **a** and **b**
The cosine of the angle $$\theta$$ between two vectors is given by:
$$
\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}
$$
First, find the magnitudes of **a** and **b**:
$$
|\mathbf{a}| = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
$$
$$
|\mathbf{b}| = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10}
$$
Now, substitute the values into the formula:
$$
\cos \theta = \frac{-3}{3\sqrt{5} \times \sqrt{10}} = \frac{-3}{3\sqrt{50}} = \frac{-1}{\sqrt{50}} = \frac{-1}{5\sqrt{2}} = \frac{-\sqrt{2}}{10}
$$
**Answer:** (D) $$-\frac{\sqrt{2}}{10}$$

### Summary of Answers:
- **12.** (C) $$-3$$
- **13.** (D) $$-\frac{\sqrt{2}}{10}$$
- **12.** (C) $$-3$$ - **13.** (D) $$-\frac{\sqrt{2}}{10}$$

If and , use the information to
answer questions 12 and 13 .
12. Find ?
13. Given that is the angle between a and , find ? (A)

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If and , use the information to answer questions 12 and 13 . 12. Find ? 13. Given that is the angle between a and , find ? (A)