Condition That 2 Lines Are Coplanar

Q.

What is Directrix in Conic Section?

Q. The lines $\overrightarrow{r}=(\hat{i}-\hat{j})+l(2\hat{i}+\hat{k})$ and $\overrightarrow{r}=(2\hat{i}-\hat{j})+m(\hat{i}+\hat{j}-\hat{k})$

1. do not intersect for any values of $l$ and $m$
2. intersect when $l=1$ and $m=2$
3. intersect when $l=2$ and $m=\frac{1}{2}$
4. intersect for all values of $l$ and $m$

Q. The value of p for which the straight lines $\overrightarrow{r}=(2\hat{i}+9\hat{j}+13\hat{k})+t(\hat{i}+2\hat{j}+3\hat{k})$ and $\overrightarrow{r}=(-3\hat{i}+7\hat{j}+p\hat{k})+s(-\hat{i}+2\hat{j}-3\hat{k})$ are coplanar is

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

Q. If the vectors $a\hat{i}+\hat{j}+\hat{k},\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar $(a\ne b\ne c\ne 1)$, then the value of $abc-(a+b+c)=$

1. $2$
2. $-2$
3. $0$
4. $-1$

Q. Each of the $n^{th}$ roots of unity are unimodular in nature.

1. False
2. True

Q. The shortest distance between the lines $\overrightarrow{r}=(4\hat{i}-\hat{j})+\lambda (\hat{i}+2\hat{j}-3\hat{k}),\lambda ϵR$ and $\overrightarrow{r}=(-\hat{i}-\hat{j}+2\hat{k})+\mu (2\hat{i}+4\hat{j}-5\hat{k}),\mu ϵR$ is

1. $\frac{6}{5}$
2. $\frac{1}{\sqrt{5}}$
3. $\frac{10}{\sqrt{5}}$
4. none of these

Q. The value of p for which the straight lines $\overrightarrow{r}=(2\hat{i}+9\hat{j}+13\hat{k})+t(\hat{i}+2\hat{j}+3\hat{k})$ and $\overrightarrow{r}=(-3\hat{i}+7\hat{j}+p\hat{k})+s(-\hat{i}+2\hat{j}-3\hat{k})$ are coplanar is

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

Q. For the given statements,
Statement $1$: Links $\overrightarrow{r}=\hat{i}-\hat{j}+\lambda (\hat{i}+\hat{j}-\hat{k})\text{and}\overrightarrow{r}=2\hat{i}-\hat{j}+\mu (\hat{i}+\hat{j}-\hat{k})$ do not intersect.
Statement $2$: Skew lines never intersect.
Which of the following is correct:

1. Both the statements are true, and Statement $2$ is the correct explanation for statement $1$.
2. Both satements are true, but statement $2$ is not the correct explanation for statement $1$.
3. Statement $1$ is true and statement $2$ is false
4. Statement $1$ is false and statement $2$ is true

Q. The shortest distance between the lines $\overrightarrow{r}=3\hat{i}+5\hat{j}+7\hat{k}+\lambda (\hat{i}+2\hat{j}+\hat{k})and\overrightarrow{r}=-\hat{i}-\hat{j}-\hat{k}+\mu (7\hat{i}-6\hat{j}+\hat{k})is$

1. $\frac{16}{5\sqrt{5}}$
2. $\frac{26}{5\sqrt{5}}$
3. $\frac{36}{5\sqrt{5}}$
4. $\frac{46}{5\sqrt{5}}$

Q. The value of p for which the straight lines $\overrightarrow{r}=(2\hat{i}+9\hat{j}+13\hat{k})+t(\hat{i}+2\hat{j}+3\hat{k})$ and $\overrightarrow{r}=(-3\hat{i}+7\hat{j}+p\hat{k})+s(-\hat{i}+2\hat{j}-3\hat{k})$ are coplanar is

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

Q. The value of p for which the straight lines $\overrightarrow{r}=(2\hat{i}+9\hat{j}+13\hat{k})+t(\hat{i}+2\hat{j}+3\hat{k})$ and $\overrightarrow{r}=(-3\hat{i}+7\hat{j}+p\hat{k})+s(-\hat{i}+2\hat{j}-3\hat{k})$ are coplanar is

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

Q. Show that the lines :

$\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (\hat{i}-\hat{j}+\hat{k})$

$\overrightarrow{r}=4\hat{j}+2\hat{k}+\mu (2\hat{i}-\hat{j}+3\hat{k})$ are coplanar.

Also, find the equation of the plane containing these lines.

Q. Shortest distance between lines
$\overrightarrow{r}=(5i+7j+3k)+\lambda (5i-16j+7k)$ and $\overrightarrow{r}=(9i+13j+15k)+\mu (3i+8j-5k):$

1. $10$unit
2. $12$unit
3. $14$unit
4. None of these