Question

If $A(2\overline{i}-\overline{j}-3\overline{k})$, $B(4\overline{i}+\overline{j}-\overline{k})$, $C(\overline{i}-3\overline{j}+2\overline{k})$ and $D(\overline{i}-\overline{j}-2\overline{k})$ then the vector equation of the plane parallel to $\overline{ABC}$ and passing through the centroid of the tetrahedron $ABCD$ is

• A. $\overline{r}=(2\overline{i}-\overline{j}-\overline{k})+s(2\overline{i}+2\overline{j}+2\overline{k})+t(\overline{i}+\overline{2j}-\overline{5k})$
• B. $\overline{r}=(2\overline{i}-\overline{j}-3\overline{k})+s(\overline{i}+\overline{j}+\overline{k})+t(\overline{i}-\overline{k})$
• C. $\overline{r}=(2\overline{i}-\overline{j}-\overline{k})+s(\overline{i}+\overline{j}+\overline{k})+t(\overline{i}+2\overline{j}-5\overline{k})$
• D. $\overline{r}=(2\overline{i}-\overline{j}-\overline{k})+s(\overline{i}+\overline{j}-\overline{k})+t(\overline{i}+2\overline{j}+5\overline{k})$

Answer 1

The correct option is A $\overline{r}=(2\overline{i}-\overline{j}-\overline{k})+s(2\overline{i}+2\overline{j}+2\overline{k})+t(\overline{i}+\overline{2j}-\overline{5k})$

$A\left(\overrightarrow{a}\right)=2\overrightarrow{i}-\overrightarrow{j}-3\overrightarrow{k},B\left(\overrightarrow{b}\right)=4\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k},C\left(\overrightarrow{b}\right)=\overrightarrow{i}-3\overrightarrow{j}+2\overrightarrow{k}$ and $D\left(\overrightarrow{b}\right)=\overrightarrow{i}-\overrightarrow{j}-2\overrightarrow{k}.$

$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}=8\hat{i}-4\hat{j}-4\hat{k}$

Centroid, $\overrightarrow{g}=\frac{\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}+\overrightarrow{d}}{4}=\frac{8\hat{i}-4\hat{j}-4\hat{k}}{4}=2\hat{i}-\hat{j}-\hat{k}$

$\overrightarrow{AB}=\overrightarrow{B}-\overrightarrow{A}=2\hat{i}+2\hat{j}+2\hat{k}$

$\overrightarrow{CA}=\overrightarrow{A}-\overrightarrow{C}=\hat{i}+2\hat{j}-5\hat{k}$

$\overrightarrow{r}-\overrightarrow{g}=s\overline{AB}+t\overline{CA}$ (vector form of plane)

$\overrightarrow{r}=\overrightarrow{g}=s\overline{AB}+t\overline{CA}$

$\therefore \overrightarrow{r}=(2\hat{i}-\hat{j}-\hat{k})+s(2\hat{i}+2\hat{j}+2\hat{k})+t(\hat{i}+2\hat{j}-5\hat{k})$Ans