Vector equations of lines and planes (6.1-6.12)

the plane $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} u + \mathbf{b} v$ in the form $(\mathbf{r}, \mathbf{n}) = D(\mathbf{a} \nparallel \mathbf{b})$

the line $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$ in the form $\left[\mathbf{r}, \mathbf{a}\right] = \mathbf{b}$ $(\mathbf{a} \neq 0)$

the line $\left[\mathbf{r}, \mathbf{a}\right] = \mathbf{b}$ in the form $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$ $(\mathbf{a} \neq 0)$

the line $\left(\mathbf{r}, \mathbf{n}_{i}\right) = D_{i}, i = 1, 2$, in the form $\left[\mathbf{r}, \mathbf{a}\right] = \mathbf{b}\left(\mathbf{n}_{1} \nparallel \mathbf{n}_{2}\right)$

the line $\left(\mathbf{r}, \mathbf{n}_{i}\right) = D_{i}, i = 1, 2$, in the form $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$ $(\mathbf{a} \neq 0)$.

intersect along a line

are parallel but do not coincide

coincide.

intersect (i.e. have a unique common point)

are skew

are parallel but do not coincide

coincide.

do they intersect (have a unique common point)

are they parallel (have no common points)

does the line lie in the plane?

the line $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$ with the plane $(\mathbf{r}, \mathbf{n}) = D$ (if $(\mathbf{a}, \mathbf{n}) \neq 0$)

the line $\left[\mathbf{r}, \mathbf{a}\right] = \mathbf{b}$ with the plane $(\mathbf{r}, \mathbf{n}) = D$ (if $(\mathbf{a}, \mathbf{n}) \neq 0)$).

the line passing through the point $M_{0}$ perpendicular to the plane $(\mathbf{r}, \mathbf{n}) = D$

the plane passing through the point $M_{0}$ perpendicular to the line $\mathbf{r} = \mathbf{r}_{1} + \mathbf{at}$.

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

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

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

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

the projection onto the plane $(\mathbf{r}, \mathbf{n}) = D$ $(\mathbf{n} \neq 0)$ of the line $\mathbf{r} = \mathbf{r}_{0} + \mathbf{a} t$ $(\mathbf{a} \neq 0)$, not perpendicular to this plane

the line meeting the line $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a} t$ $(\mathbf{a} \neq 0)$ at a right angle and passing through the point $M_{0}\left(\mathbf{r}_{0}\right)$, not lying on the given line (the perpendicular dropped from the point $M_{0}\left(\mathbf{r}_{0}\right)$ onto the line $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a} t$)

the line meeting the two skew lines $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a}_{1} t$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a}_{2} t$ $(\mathbf{a} \neq 0)$ and passing through the point $M_{0}\left(\mathbf{r}_{0}\right)$, not lying on either of these lines

the line meeting the two skew lines $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a}_{1} t$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a}_{2} t$ $(\mathbf{a} \neq 0)$ at right angles (the common perpendicular to these lines).

from the point $M_{0}\left(\mathbf{r}_{0}\right)$ to the plane $(\mathbf{r}, \mathbf{n}) = D$ $(\mathbf{n} \neq 0)$

between the two parallel planes $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a} u + \mathbf{b} v$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a} u + \mathbf{b} v$ $(\mathbf{a} \neq 0)$

between the two parallel planes $(\mathbf{r}, \mathbf{n}) = D_{1}$ and $(\mathbf{r}, \mathbf{n}) = D_{2}$ $(\mathbf{n} \neq 0)$

from the point $M_{0}\left(\mathbf{r}_{0}\right)$ to the line $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a} t$ $(\mathbf{a} \neq 0)$

from the point $M_{0}\left(\mathbf{r}_{0}\right)$ to the line $\left[\mathbf{r}, \mathbf{a}\right] = \mathbf{b}$ $(\mathbf{a} \neq 0)$

between the two parallel lines $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a} t$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a} t$ $(\mathbf{a} \neq 0)$

between the two parallel lines $\left[\mathbf{r}, \mathbf{a}\right] = \mathbf{b}_{1}$ and $\left[\mathbf{r}, \mathbf{a}\right] = \mathbf{b}_{2}$ $(\mathbf{a} \neq 0)$

between the two skew lines $\mathbf{r} = \mathbf{r}_{1} + \mathbf{a}_{1} t$ and $\mathbf{r} = \mathbf{r}_{2} + \mathbf{a}_{2} t$ $(\mathbf{a} \neq 0)$

between the two skew lines $\left[\mathbf{r}, \mathbf{a}_{1}\right] = \mathbf{b}_{1}$ and $\left[\mathbf{r}, \mathbf{a}_{2}\right] = \mathbf{b}_{2}$ $(\mathbf{a} \neq 0)$.