IB Mathematics Applications & Interpretation 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 $\mathbf{d}=\left(\begin{array}{c}3\\ -5\\ 2\end{array}\right)$.

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 $\frac{x-1}{2}=\frac{y+3}{-4}=\frac{z-2}{3}$ in parametric form.

Express the line given in vector form $\mathbf{r}=\left(\begin{array}{c}0\\ 1\\ 6\end{array}\right)+k\left(\begin{array}{c}4\\ 2\\ 1\end{array}\right)$ 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 $\mathbf{r}=\left(\begin{array}{c}3\\ -2\\ 5\end{array}\right)+t\left(\begin{array}{c}-1\\ 4\\ 2\end{array}\right)$ into its symmetric form.

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