Question
If $\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle, \mathbf{r}=\langle x, y\rangle,$ and $c>0,$ describe the set of allpoints $P(x, y)$ such that $\|\mathbf{r}-\mathbf{a}\|=c$.
Step 1
Step 1: Given vectors $\mathbf{a}=\left\langle a_{1}, a_{2}\right\rangle$ and $\mathbf{r}=\langle x, y\rangle$, we can express the vector $\mathbf{r}-\mathbf{a}$ as $\langle x-a_{1}, y-a_{2}\rangle$. Show more…
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$$\begin{array}{l}{\text { If } \mathbf{r}=[x, y], \mathbf{r}_{1}=\left[x_{1}, y_{1}\right], \text { and } \mathbf{r}_{2}=\left[x_{2}, y_{2}\right], \text { describe the }} \\ {\text { set of all points }(x, y) \text { such that }\left|\mathbf{r}-\mathbf{r}_{1}\right|+\left|\mathbf{r}-\mathbf{r}_{2}\right|=k} \\ {\text { where } k>\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right|}\end{array}$$
Vectors and Matrix Models
Vectors
Let $\mathbf{v}_{0}=\langle a, b\rangle .$ Describe the set of points $(x, y)$ such that, for $\mathbf{v}=\langle x, y\rangle$ (a) $\mathbf{v} \cdot \mathbf{v}_{0}=0$. (b) $\left(\mathbf{v}-\mathbf{v}_{0}\right) \cdot \mathbf{v}_{0}=0$. (c) $\left(\mathbf{v}-\mathbf{v}_{0}\right) \cdot \mathbf{v}=0$
Dot Product
If $ r = \langle x, y \rangle , r_1 = \langle x_1, y_1 \rangle $, and $ r_2 = \langle x_2, y_2 \rangle $, describe the set of all points $ (x, y) $ such that $ \mid r - r_1 \mid + \mid r - r_2 \mid = k $, where $ k > \mid r_1 - r_2 \mid $.
Vectors and the Geometry of Space
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