Question Type 3: Converting between different forms of a line Exercises

Find the Cartesian form of the line passing through point $P(2, -1, 3)$ with direction vector $\mathbf{d} = \begin{pmatrix} 3 \\ -5 \\ 2 \end{pmatrix}$.

Find parametric equations for the line through $A(0,2)$ and $B(3,-1)$ in the plane.

Convert the vector equation $\mathbf{r}=(1,0,-2)+\lambda(2,-3,5)$ into symmetric equations.

Determine parametric equations for the line of intersection of the planes $3x-y+2z=7$ and $x+2y-z=3$.

Find the parametric and symmetric equations of the line through the origin perpendicular to both vectors $(1, 2, 3)$ and $(4, -1, 2)$.

Express the line defined by $\displaystyle \frac{x-1}{2}=\frac{y+3}{-4}=\frac{z-2}{3}$ in parametric form.

Express the line given in vector form $\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 6 \end{pmatrix} + k \begin{pmatrix} 4 \\ 2 \\ 1 \end{pmatrix}$ in parametric equations.

A line passes through $A(1,2,3)$ and $B(4,0,6)$. Write its Cartesian (symmetric) equations.

Find the symmetric form of the line through $P(2,-1,4)$ parallel to the intersection of the planes $x-y+z=2$ and $2x+y-3z=5$.

Express the line $y=\frac{2}{3}x+5$ in vector form in $\mathbb{R}^3$ by setting $z=0$.

Convert the vector equation $\mathbf{r} = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix} + t \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix}$ into its symmetric form.

Convert the parametric equations $x = 1 + 2t$ and $y = 4 - 3t$ into a Cartesian equation in $x$ and $y$.