Vector equation of a Line and Related Concepts

The magnitudes of vectors  $\overrightarrow{A},\overrightarrow{B}\mathrm{and}\overrightarrow{C}$  are $3,$ $4$ and $5$ units respectively. If  $\overrightarrow{A}+\overrightarrow{B}=\overrightarrow{C}$ , then the angle between  $\overrightarrow{A}\mathrm{and}\overrightarrow{B}$  is :

The position vectors of two points $A$ and $B$ are $\hat{i}-\hat{j}$ and $\hat{j}+\hat{k}$ respectively.

Consider the following points :

$1.$ $\left(-1,-3, 1\right)$

$2.$ $\left(-1, 3, 2\right)$

$3.$ $\left(-2, 5, 3\right)$

Which of the above points lie on the line joining $A$ and $B$?

The equation of a line passing through the point $\left(2,4,6\right)$ and parallel to the line $3x+4=4y-1=1-4z$ is

If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha }$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta }$ intersect, then the magnitude of the minimum value of $8\alpha \beta$ is _____.

If the two lines $\frac{x-2m}{2m+5}=\frac{y}{8m}=\frac{z-4}{2}$ and $\frac{x-2}{m-2}=\frac{y}{-1}=\frac{z-2m}{1-3m}$ are parallel for some $m\in R$, then the distance between them is:

What is the condition to check whether two vectors are intersecting or not $?$

Find the value of $k$ so that the lines $x=-y=kz$ and $x-2=2y+1=-z+1$ are perpendicular to each other. Also, find the equation of line passing through $\left(1,1,1\right)$ and perpendicular to these lines.

Write the parametric equations of the line which passes through $\left(1,2,-3\right)$ and $\left(3,3,2\right)$. If $y=\sin x$ then find $\frac{dy}{dx}.$

In a realistic city model, a metro track and a road above it are straight lines represented by $\hat{i}-2\hat{j}+2\hat{k}+\lambda \left(\hat{i}+2j-3\hat{k}\right)$ and $2\hat{i}-2\hat{j}+3\hat{k}+\mu \left(-\hat{i}+\hat{j}+2\hat{k}\right)$ respectively.

Find the shortest distance between the metro track and the road, in the model. Show your work.

Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are

$\overrightarrow{r}=\hat{i}+\hat{j}+\lambda \left(2\hat{i}-\hat{j}+\hat{k}\right)$

$\overrightarrow{r}=\widehat{2i}+\hat{j}-\hat{k}+\mu \left(3\hat{i}-\widehat{5j}+2\hat{k}\right)$

The point $\mathrm{A}(1,2,3), \mathrm{B}(-1,-2,-1) \mathrm{and} \mathrm{C}(2,3,2)$ are three vertices of a parallelogram $\mathrm{ABCD}$ then find the equation of $\mathrm{CD}$

If line $\frac{x-1}{1}=\frac{y}{-1}=\frac{z+2}{2}$ lies completely in the plane $2x+3y+\lambda z+k=0$, then-

  Consider the points $A(0,1) and B(2,0)$, and point P on the line 4x+3y+9=0. The coordinates of P such that $\left|PA-PB\right|$ is maximum are