question_answer

            Considering the Earth as a plane having equation \[\vec{r}(2\hat{i}+2\hat{k})=8\] and the Mars as a line having equation \[\vec{r}=\hat{i}-2\hat{j}+\hat{k}+\lambda (2\hat{i}+3\hat{j}+\hat{k})\] such that their common point is \[(\alpha ,\,\,\beta ,\,\,\gamma ).A\] monument is standing vertically on the Earth such that its peak is at the point \[(-\,2,\,\,-\,6,\,\,-\,11).\] (i) Find the distance between the peak and common point of Earth and Mars. (ii) How can we save our monuments?

Answer:

(i) Given equation of the line (i.e. Mars) in vector form is                         \[\vec{r}=\hat{i}-2\hat{j}+\hat{k}+\lambda (2\hat{i}+3\hat{j}+\hat{k})\]                         Then, equation of line in cartesian form is             \[\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{1}=k\] (say) \[\Rightarrow \]   \[x=2k+1,\] \[y=3k-2,\] \[z=k+1\] The values of x, y and z, so obtained are the general coordinates of any point say A on the line, i.e. coordinates of any point A on the line are             \[(2k+1,\,3k-2,\,k+1)\]                ?(i) Also, the given equation of the plane (i.e. Earth) in vector form is             \[\vec{r}\cdot (2\hat{i}+2\hat{k})=8\] or \[\vec{r}\cdot (\hat{i}+\hat{k})=4\]        ?(ii) Then, equation of plane in cartesian form is             \[x+z=4\]\[[\text{put}\,\,\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}]\]    ?(ii) Let common point A be the intersection point of line and plane, so the coordinates of A will satisfy the equation of plane. \[\therefore \]      \[3k+2+2k+2=4\] \[\Rightarrow \]   k = 0    On putting k = 0 in coordinates of A we get             \[A=(1,\,\,-2,\,\,1)\] Now, distance of the given point \[P\,(-2,\,\,-6,\,\,-11)\]from \[A\,(1,\,\,-2,\,\,1)\] is             \[AP=\sqrt{{{(1+2)}^{2}}+{{(-2+6)}^{2}}+{{(1+11)}^{2}}}\]             [by distance formula between two points]             \[=\sqrt{{{3}^{2}}+{{4}^{2}}+{{(12)}^{2}}}\]             \[=\sqrt{9+16+144}\]\[=\sqrt{169}\]= 13 units (ii)         (a) We should not harm any monument. (b) We should not write anything on them. (c) We should respect our national heritage.