Shortest Distance between Two Lines

Q. Let $A\left(\overrightarrow{a}\right),$ $B\left(\overrightarrow{b}\right),$ $C\left(\overrightarrow{c}\right)$ and $D\left(\overrightarrow{d}\right)$ be four points such that $\overrightarrow{a}=-2\hat{i}+4\hat{j}+3\hat{k},$ $\overrightarrow{b}=2\hat{i}-8\hat{j},$ $\overrightarrow{c}=\hat{i}-3\hat{j}+5\hat{k},$ $\overrightarrow{d}=4\hat{i}+\hat{j}-7\hat{k}.$ If $m$ is the shortest distance between the lines $AB$ and $CD,$ then which of the following is (are) CORRECT?

1. $m=0,$ hence $AB$ and $CD$ intersect
2. $m=\frac{\left[\overrightarrow{AB}\overrightarrow{CD}\overrightarrow{BD}\right]}{|\overrightarrow{AB}\times \overrightarrow{CD}|}$
3. $AB$ and $CD$ are skew lines and $m=\frac{23}{13}$
4. $m=\frac{\left[\overrightarrow{AB}\overrightarrow{CD}\overrightarrow{AC}\right]}{|\overrightarrow{AB}\times \overrightarrow{CD}|}$

Q. How to prepare straight lines , pair of straight lines, locus and transformation of Axes till advanced level

Q. If the shortest distance between the lines $\overrightarrow{r_1}=\alpha \hat{i}+2\hat{j}+2\hat{k}+\lambda (\hat{i}-2\hat{j}+2\hat{k}),\lambda \in R,\alpha >0$ and $\overrightarrow{r_2}=-4\hat{i}-\hat{k}+\mu (3\hat{i}-2\hat{j}-2\hat{k}),\mu \in R$ is $9,$ then $\alpha$ is equal to

Q. Minimise and Maximise $Z=x+2y$
subject to $x+2y\ge 100,2x-y\le 0,2x+y\le 200;x,y\ge 0$

Q. Maximize $Z=3x+4y$
subject to the constraints: $x+y\le 4,x\ge 0,y\ge 0$

Q. Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}:a,b,c\in \{-1,1\left\}\right\}$. Three non-coplanar vectors can be chosen from $V$ in $2^p$ ways. Then $p$ is

Q. Find the shortest distance between $\overline{r}=\hat{i}+\lambda (\hat{j}+\hat{k})and\overline{r}=2\hat{j}+\beta \left(2\hat{k}\right)$.

1. 3
2. 4
3. 1
4. 2

Q. Minimise $Z=x+2y$
subject to $2x+y\ge 3,x+2y\ge 6,x,y\ge 0$

Q. If the vectors $3\hat{i}+\hat{j}-2\hat{k}$, $\hat{i}-3\hat{j}+4\hat{k}$ are is adjacent sides are $3\hat{i}+\hat{j}-2\hat{k}$, $\hat{i}-3\hat{j}+4\hat{k}$ diagonals of a quadrilateral, then the vector area

1. $\hat{i}+7\hat{j}-5\hat{k}$
2. $\hat{i}-7\hat{j}+5\hat{k}$
3. $-\hat{i}+2\hat{j}+5\hat{k}$
4. $-\hat{i}-7\hat{j}-5\hat{k}$

Q. Find the projection of $\hat{i}+3\hat{j}+j\hat{k}$ on the vector $7\hat{i}-\hat{j}+8\hat{k}$.

Q. The correct statement about the lines $\overrightarrow{r}=\hat{i}+2\hat{j}+s(3\hat{i}-\hat{j}+\hat{k})$ and $\overrightarrow{r}=10\hat{i}-\hat{j}+\alpha \hat{k}+t(-6\hat{i}+2\hat{j}-2\hat{k})$ is/are:

1. They are skew lines for any value of $\alpha$
2. They are coincident for $\alpha =3$
3. They are parallel for $\alpha =2$
4. They have more than one common point for $\alpha =1$

Q. The shortest distance (in units) between the lines whose equations are $\overrightarrow{r}=\hat{i}+2\hat{j}+7\hat{k}+\lambda (\hat{i}+2\hat{j}+3\hat{k})$ and $\overrightarrow{r}=-\hat{i}-2\hat{j}+3\hat{k}+s(7\hat{i}+6\hat{j}+3\hat{k})$ is:

1. $\frac{30}{\sqrt{133}}$
2. $\frac{8}{\sqrt{133}}$
3. $\frac{40}{\sqrt{113}}$
4. $\frac{30}{\sqrt{113}}$

Q. The circum radius of the triangle formed by the points $(0,0,0)$, $(0,0,12)$ and $(3,4,0)$ is

1. $13$
2. $\sqrt{156}$
3. $\frac{13}{2}$
4. $8$

Q. Find the projection of the vector $\hat{i}+3\hat{j}+7\hat{k}$ on the vector $7\hat{i}-\hat{j}+8\hat{k}$ .