Vector - straight lines

**geton** · November 12th 2008, 07:56 AM

The straight lines $l_1$ and $l_2$ have vector equations r = (i + 4j + 2k) + t(8i + 5j + k) and r = (i + 4j + 2k) + s(3i + j) respectively. P is the point with coordinate (1, 4, 2) & Q is the point with coordinate (9, 9, 3).

Find the possible coordinates of the point R, such that R lies on $l_2$ and PQ = PR.

How could I do this?

**Soroban** · November 12th 2008, 09:35 AM

Hello, geton!

Are there more parts to this problem?
There is too much information . . .

The line $L$ has vector equation: . $r \;=\;(i + 4j + 2k) + s(3i + j)$

$P$ is the point $(1, 4, 2)$ and $Q$ is the point $(9, 9, 3).$

Find the possible coordinates of the point $R$ such that $R$ lies on $L$ and $PQ = PR.$

Distance $PQ$ is: . $\sqrt{(9-1)^2 + (9-4)^2 + (3-2)^2} \;=\; \sqrt{8^2+ 5^2 + 1^2} \:=\:\sqrt{90}$ .[1]

Line $L$ has parametric equations: . $\begin{array}{c}x \:=\:1 + 3s \\ y \:=\:4 + s \\ z \:=\:2 \end{array}$ .(a)

Point $R$ has coordinates: . $(1+3s,\:4+s,\:2)$

Distance $QR$ is: . $\sqrt{(1+3s-4)^2 + (4+s-9)^2 + (2-3)^2} \;=\;\sqrt{10s^2 - 58s + 90}$ .[2]

Equate [2] and [1]: . $\sqrt{10s^2 - 58s + 90} \;=\;\sqrt{90} \quad\Rightarrow\quad 10s^2 - 58s + 90 \;=\;90$

. . $10s^2 - 58s \:=\:0 \quad\Rightarrow\quad 2s(5s-29) \:=\:0 \quad\Rightarrow\quad s \;=\;0,\:\tfrac{29}{5}$

Substitute into (a): . $\begin{array}{c|c}<br /> s=0 & s = \frac{29}{5} \\ \\[-4mm] \hline \\[-3mm] x = 1 & x = \frac{92}{5} \\ \\[-3mm] y = 4 & y = \frac{49}{5} \\ \\[-4mm] z = 2 & z = 2 \end{array}$

Therefore: . $R \;=\;(1,4,2)\:\text{ or }\:\left(\tfrac{92}{5},\:\tfrac{49}{5},\:2\right)$

**geton** · November 12th 2008, 10:44 AM

Originally Posted by Soroban

Hello, geton!

Are there more parts to this problem?
There is too much information . . .

Yes, I tried to short the question. But ...

Distance $PQ$ is: . $\sqrt{(9-1)^2 + (9-4)^2 + (3-2)^2} \;=\; \sqrt{8^2+ 5^2 + 1^2} \:=\:\sqrt{90}$ .[1]

Line $L$ has parametric equations: . $\begin{array}{c}x \:=\:1 + 3s \\ y \:=\:4 + s \\ z \:=\:2 \end{array}$ .(a)

Point $R$ has coordinates: . $(1+3s,\:4+s,\:2)$

Distance $QR$ is: . $\sqrt{(1+3s-4)^2 + (4+s-9)^2 + (2-3)^2} \;=\;\sqrt{10s^2 - 58s + 90}$ .[2]

Equate [2] and [1]: . $\sqrt{10s^2 - 58s + 90} \;=\;\sqrt{90} \quad\Rightarrow\quad 10s^2 - 58s + 90 \;=\;90$

. . $10s^2 - 58s \:=\:0 \quad\Rightarrow\quad 2s(5s-29) \:=\:0 \quad\Rightarrow\quad s \;=\;0,\:\tfrac{29}{5}$

Substitute into (a): . $\begin{array}{c|c}<br /> s=0 & s = \frac{29}{5} \\ \\[-4mm] \hline \\[-3mm] x = 1 & x = \frac{92}{5} \\ \\[-3mm] y = 4 & y = \frac{49}{5} \\ \\[-4mm] z = 2 & z = 2 \end{array}$

Therefore: . $R \;=\;(1,4,2)\:\text{ or }\:\left(\tfrac{92}{5},\:\tfrac{49}{5},\:2\right)$

Thank you so much for giving the method to do. But the distance QR will be PR.

Thanks a lot.

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