Find the equation of plane determined |
by points \[A(3,\,\,-\,1,\,\,2),\] B(5, 2, 4), \[C(-\,1,\,\,-\,1,\,\,6)\] and hence find the distance between plane and point P(6, 5, 9). |
OR |
Show that the lines |
\[\vec{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})\] and |
\[\vec{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})\] are intersecting. |
Hence, find their point of intersection. |
Answer:
Given points are \[A\,(3,\,\,-1,\,\,2),\]\[B\,(5,\,\,2,\,\,4)\]and\[C\,(-\,1,\,\,-\,1,\,\,6)\]. Now, equation of plane passing through A, B and C is given by \[\Rightarrow \] \[\Rightarrow \] Expanding along \[{{R}_{1}}\]we get \[(x-3)\,\,(12-0)-(y+1)\,\,(8+8)+(z-2)\,\,(0+12)=0\]\[\Rightarrow \] \[12x-36-16y-16+12z-24=0\] \[\Rightarrow \] \[12x-16y+12z=76\] \[\Rightarrow \] \[3x-4y+3z=19\] Now, distance of the point (6, 5, 9) from the plane (i) is \[d=\left| \frac{3\,(6)-4\,(5)+3\,(9)-19}{\sqrt{{{3}^{2}}+{{4}^{2}}+{{3}^{2}}}} \right|\] \[=\left| \frac{18-20+27-19}{\sqrt{9+16+9}} \right|\] \[=\left| \frac{6}{\sqrt{34}} \right|\] \[=\frac{6}{\sqrt{34}}units\] Or Consider the lines \[\overrightarrow{r}=3\,\hat{i}+2\,\hat{i}-4\hat{k}+\lambda \,(\hat{i}+2\hat{j}+2\hat{k})\] ?(i) and \[\overrightarrow{r}=5\,\hat{i}-2\hat{j}+\mu \,(3\,\hat{i}+2\hat{j}+6\hat{k})\] ?(ii) Here, \[{{\overrightarrow{a}}_{1}}=3\,\hat{i}-2\hat{j}-4k,\,\,{{\overrightarrow{b}}_{1}}=\hat{i}+2\hat{j}+2\hat{k}\] and \[{{\overrightarrow{a}}_{2}}=5\,\hat{i}-2\hat{j},\,\,{{\overrightarrow{b}}_{2}}=3\,\hat{i}+2\hat{j}+6\hat{k}\] \[\therefore \] \[{{\overrightarrow{a}}_{2}}-{{\overrightarrow{a}}_{1}}=5\,\hat{i}-2\hat{j}-3\,\hat{i}-2\hat{j}+4\hat{k}\] \[=2\hat{i}-4\hat{j}+4\hat{k}\] and Consider\[({{\overrightarrow{a}}_{2}}-{{\overrightarrow{a}}_{1}})\cdot ({{\overrightarrow{b}}_{1}}\times {{\overrightarrow{b}}_{2}})=16-16=0\] As \[({{\overrightarrow{a}}_{2}}-{{\overrightarrow{a}}_{1}})\cdot ({{\overrightarrow{b}}_{1}}\times {{\overrightarrow{b}}_{2}})=0\] \[\Rightarrow \]Lines are intersecting (coplanar and \[{{\overrightarrow{b}}_{1}}\ne {{\overrightarrow{b}}_{2}}\]) General point on line (i) is \[\overrightarrow{r}=(3+\lambda )\,\hat{i}+(2+2\lambda )\hat{j}+(-\,4+2\lambda )\hat{k}\] ?(iii) General point on line (ii) is \[\overrightarrow{r}=(5+3\mu )\,\hat{i}+(-\,2+2\mu )\hat{j}+6\mu \hat{k}\] ?(iv) If Eqs. (iii) and (iv) represent same point \[3+\lambda =5+3\mu \Rightarrow \lambda -3\mu =2\] ?(v) \[2+2\lambda =-\,2+2\mu \] \[\Rightarrow \] \[2\lambda -2\mu =-\,4\] and \[-\,4+2\lambda =6\mu \] \[\Rightarrow \] \[2\lambda -6\mu =4\] ?(vi) On solving Eqs. (vi) and (vii), we get \[\lambda =-\,4,\]\[\mu =-\,2\] On putting the values of \[\lambda \] and \[\mu \] in Eq. (v), we get \[-\,4+6=2\] [true] Hence, for \[\lambda =-\,4,\]\[\mu =-\,2,\]Eqs. (iii) and (iv) represent the same point. Position vector of point of intersection is\[\overrightarrow{r}=-\,\hat{i}-6\hat{j}-12\hat{k}\] \[\therefore \]Point of intersection is \[(-\,1,\,\,-\,6,\,\,-\,12)\]
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