Write the vector equation of the line (x-2)/(3)=tfrac…

Write the vector equation of the line $\tfrac{x-2}{3}=\tfrac y5=\tfrac{z+5}{2}$.

Step 1: Solution

Here $\vec r_0=\{2,0,-5\}$ and $\vec d=\{3,5,2\}$: \[\vec c=\vec r_0\times\vec d=\{0\cdot2-(-5)\cdot5,\ (-5)\cdot3-2\cdot2,\ 2\cdot5-0\cdot3\}=\{25,-19,10\}.\]

\[\vec r\times(3\vec i+5\vec j+2\vec k)=25\vec i-19\vec j+10\vec k\]

Check: $\vec c\cdot\vec d=75-95+20=0$. Using any other point of the line gives the same $\vec c$, since $(\vec r_0+t\vec d)\times\vec d=\vec r_0\times\vec d$.