Question

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the
(i) yz-plane
(ii) zx-plane

Answer 1

$$\begin{gathered} \left(i\right)\text{The equation of the line through the points (5, 1, 6) and (3, 4, 1) is} \\ \frac{x-5}{3-5}=\frac{y-1}{4-1}=\frac{z-6}{1-6} \\ \Rightarrow \frac{x-5}{-2}=\frac{y-1}{3}=\frac{z-6}{-5} \\ \text{The coordinates of any point on this line are of the form} \\ \frac{x-5}{-2}=\frac{y-1}{3}=\frac{z-6}{-5}=\lambda \\ \Rightarrow x=-2\lambda +5;y=3\lambda +1;z=-5\lambda +6 \\ \text{So, the coordinates of the point on the given line are}\left(-2\lambda +5,3\lambda +1,-5\lambda +6\right)\text{.} \\ \text{Since this point lies on the}\mathit{\text{YZ}}\text{- plane,} \\ x=0 \\ \Rightarrow -2\lambda +5=0 \\ \Rightarrow \lambda =\frac{5}{2} \\ \mathrm{So,\; the\; coordinates\; of\; the\; point}\text{are} \\ \left(-2\lambda +5,3\lambda +1,-5\lambda +6\right) \\ =\left(-2\left(\frac{5}{2}\right)+5,3\left(\frac{5}{2}\right)+1,-5\left(\frac{5}{2}\right)+6\right) \\ =\left(0,\frac{17}{2},\frac{13}{2}\right) \end{gathered}$$

$$\begin{gathered} \left(ii\right)\text{The equation of the line through the points (5, 1, 6) and (3, 4, 1) is} \\ \frac{x-5}{3-5}=\frac{y-1}{4-1}=\frac{z-6}{1-6} \\ \Rightarrow \frac{x-5}{-2}=\frac{y-1}{3}=\frac{z-6}{-5} \\ \text{The coordinates of any point on this line are of the form} \\ \frac{x-5}{-2}=\frac{y-1}{3}=\frac{z-6}{-5}=\lambda \\ \Rightarrow x=-2\lambda +5;y=3\lambda +1;z=-5\lambda +6 \\ \text{So, the coordinates of the point on the given line are}\left(-2\lambda +5,3\lambda +1,-5\lambda +6\right)\text{.} \\ \text{Since this point lies on the}\mathit{\text{ZX}}\text{- plane,} \\ y=0 \\ \Rightarrow 3\lambda +1=0 \\ \Rightarrow \lambda =\frac{-1}{3} \\ \mathrm{So,\; the\; coordi}\text{na}\mathrm{tes\; of\; the\; point}\text{are} \\ \left(-2\lambda +5,3\lambda +1,-5\lambda +6\right) \\ =\left(-2\left(\frac{-1}{3}\right)+5,3\left(\frac{-1}{3}\right)+1,-5\left(\frac{-1}{3}\right)+6\right) \\ =\left(\frac{17}{3},0,\frac{23}{3}\right) \end{gathered}$$