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

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

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

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

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

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

Write the line passing through $P(2,-1,3)$ with direction vector $\mathbf{d}=(3,-5,2)$ in Cartesian (symmetric) form.

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

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

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

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

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