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Let $L$ be the line $2 x+4 y=8$ and let $P$ be the point $(3,4)$ (a) Show that the points $Q(0,2)$ and $R(2,1)$ lie on $L$. (b) Let $\mathbf{u}=\overrightarrow{Q P}...

Question

Let $L$ be the line $2 x+4 y=8$ and let $P$ be the point $(3,4)$ (a) Show that the points $Q(0,2)$ and $R(2,1)$ lie on $L$. (b) Let $\mathbf{u}=\overrightarrow{Q P}$ and $\mathbf{v}=\overrightarrow{Q R},$ as shown in the figure. Find $\mathbf{w}=\operatorname{proj}_{\mathbf{v}} \mathbf{u}$. (c) Sketch a graph that explains why $|\mathbf{u}-\mathbf{w}|$ is the distance from $P$ to $L$. Find this distance. (d) Write a short paragraph describing the steps you would take to find the distance from a

Let $L$ be the line $2 x+4 y=8$ and let $P$ be the point $(3,4)$ (a) Show that the points $Q(0,2)$ and $R(2,1)$ lie on $L$. (b) Let $\mathbf{u}=\overrightarrow{Q P}$ and $\mathbf{v}=\overrightarrow{Q R},$ as shown in the figure. Find $\mathbf{w}=\operatorname{proj}_{\mathbf{v}} \mathbf{u}$. (c) Sketch a graph that explains why $|\mathbf{u}-\mathbf{w}|$ is the distance from $P$ to $L$. Find this distance. (d) Write a short paragraph describing the steps you would take to find the distance from a given point to a given line. CAN'T COPY THE FIGURE



Answers

(a) Let $ P $ be a point not on the line $ L $ that passes through the points $ Q $ and $ R $. Show that the distance $ d $ from the point $ P $ to the line $ L $ is
$$ d = \frac{\mid a \times b \mid}{\mid a \mid} $$
where $ a = \vec{QR} $ and $ b = \vec{QP} $
(b) Use the formula in part (a) to find the distance from the point $ P (1, 1, 1) $ to the line through $ Q (0, 6, 8) $ and $ R (-1, 4, 7) $


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