complete the following for each pair of vectors. do not round any intermediate computations. round your…

Explanation:

Step1: Recall the dot - product formula

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 $\mathbf{a}\cdot\mathbf{b}=a_1b_1 + a_2b_2$, and $\mathbf{a}\cdot\mathbf{b}=\vert\mathbf{a}\vert\vert\mathbf{b}\vert\cos\theta$, where $\theta$ is the angle between the two vectors, and $\vert\mathbf{a}\vert=\sqrt{a_1^{2}+a_2^{2}}$, $\vert\mathbf{b}\vert=\sqrt{b_1^{2}+b_2^{2}}$.

Step2: Calculate for $\mathbf{v}=-3\mathbf{i}+\mathbf{j}$ and $\mathbf{w}=-\mathbf{i}-3\mathbf{j}$

Calculate the dot - product

$\mathbf{v}\cdot\mathbf{w}=(-3)\times(-1)+1\times(-3)=3 - 3=0$.

Calculate the magnitudes

$\vert\mathbf{v}\vert=\sqrt{(-3)^{2}+1^{2}}=\sqrt{9 + 1}=\sqrt{10}$, $\vert\mathbf{w}\vert=\sqrt{(-1)^{2}+(-3)^{2}}=\sqrt{1 + 9}=\sqrt{10}$.

Find the angle

Since $\mathbf{v}\cdot\mathbf{w}=\vert\mathbf{v}\vert\vert\mathbf{w}\vert\cos\theta$, then $0=\sqrt{10}\times\sqrt{10}\cos\theta$, so $\cos\theta = 0$. Thus, $\theta = 90^{\circ}$.

Determine the relationship

Since $\mathbf{v}\cdot\mathbf{w}=0$, $\mathbf{v}$ and $\mathbf{w}$ are orthogonal.

Step3: Calculate for $\mathbf{u}=5\mathbf{i}-4\mathbf{j}$ and $\mathbf{r}=6\mathbf{i}+5\mathbf{j}$

Calculate the dot - product

$\mathbf{u}\cdot\mathbf{r}=5\times6+(-4)\times5=30-20 = 10$.

Calculate the magnitudes

$\vert\mathbf{u}\vert=\sqrt{5^{2}+(-4)^{2}}=\sqrt{25 + 16}=\sqrt{41}$, $\vert\mathbf{r}\vert=\sqrt{6^{2}+5^{2}}=\sqrt{36+25}=\sqrt{61}$.

Find the angle

$\cos\theta=\frac{\mathbf{u}\cdot\mathbf{r}}{\vert\mathbf{u}\vert\vert\mathbf{r}\vert}=\frac{10}{\sqrt{41}\times\sqrt{61}}\approx\frac{10}{\sqrt{2501}}\approx\frac{10}{50.01}\approx0.2$. Then $\theta=\cos^{-1}(0.2)\approx79^{\circ}$.

Determine the relationship

Since $\mathbf{u}\cdot\mathbf{r}\neq0$ and there is no scalar $k$ such that $\mathbf{u}=k\mathbf{r}$ (i.e., $\frac{5}{6}\neq\frac{-4}{5}$), $\mathbf{u}$ and $\mathbf{r}$ are neither parallel nor orthogonal.

Answer:

1. (a) $90$ (b) orthogonal
2. (a) $79$ (b) neither