Math Problems Vault

Problem 5.1

(1)(2)(3)

Under what necessary and sufficient condition do the lines $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a}_{1} t$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a}_{2} t\quad \left(\mathbf{a}_{i} \neq 0, i = 1, 2\right)$ :

?

(1)

intersect at a single point

(2)

are parallel but do not coincide

(3)

coincide?

Problem 5.2

(1)(2)

Find the angle between the lines given by their equations:

?

(1)

$\mathbf{r} = \mathbf{r}_{1} + \mathbf{a}_{1} t$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a}_{2} t\quad \left(\mathbf{a}_{i} \neq 0, i = 1, 2\right)$

(2)

$\left(\mathbf{r}, \mathbf{n}_{1}\right) = D_{1}$ and $\left(\mathbf{r}, \mathbf{n}_{2}\right) = D_{2}\left(\mathbf{n}_{1} \nparallel \mathbf{n}_{2}\right)$ .

Problem 5.3

Two lines are given by the vector equations $(\mathbf{r}, \mathbf{n}) = D$ and $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$ , with $(\mathbf{a}, \mathbf{n}) \neq 0$ . Find the position vector of the point where the lines meet.

?

Problem 5.4

(1)(2)

Given a point $M_{0}$ with position vector $\mathbf{r}_{0}$ and a line $(\mathbf{r}, \mathbf{n}) = D$ $(\mathbf{n} \neq 0)$ . Find the position vectors of:

?

(1)

the projection of point $M_{0}$ onto the line

(2)

the point $M_{1}$ symmetric to $M_{0}$ with respect to the given line.

Problem 5.5

(1)(2)

Find the distance from point $M_{0}\left(\mathbf{r}_{0}\right)$ to the line given by the equation:

?

(1)

$(\mathbf{r}, \mathbf{n}) = D$ $(\mathbf{n} \neq 0)$

(2)

$\mathbf{r} = \mathbf{r}_{1} + \mathbf{a} t$ $(\mathbf{a} \neq 0)$