Equations of Planes — Question 1

Question 1

Find scalar and vector equations of the plane through $P_0=(2,-1,3)$ with normal vector $\vec n=\left\langle 4,2,-1\right\rangle$. Find its intercepts with the coordinate axes.

Question 1 – Solution

Strategy Use $\vec n\cdot(\vec r-\vec r_0)=0$, then set two coordinates equal to zero for each intercept.

Equations The vector equation is $\left\langle 4,2,-1\right\rangle\cdot\left\langle x-2,y+1,z-3\right\rangle=0$, which simplifies to $\boxed{4x+2y-z=3}$.

Intercepts The $x$-, $y$-, and $z$-intercepts are $$\boxed{(3/4,0,0),\quad(0,3/2,0),\quad(0,0,-3)}.$$

Verification Substituting $P_0$ gives $8-2-3=3$, and each intercept satisfies the scalar equation.