Question

write the equation of the line that passes through the points (2,1,3) and (1,4,-3). write all three forms of the equations of the line.

Answer 1

There is three major forms of linear equations:

• vector form: $(x,y,z)=(x{\scriptscriptstyle 0},y{\scriptscriptstyle 0},z{\scriptscriptstyle 0})+t(a,b,c)$
• parametric form: $x=x{\scriptscriptstyle 0}+ta, y = y{\scriptscriptstyle 0}+tb,z=z{\scriptscriptstyle 0}+tc$
• symmetric form: ${\frac {x-x{\scriptscriptstyle 0}} a}={\frac {y-y{\scriptscriptstyle 0}} b}={\frac {z-z{\scriptscriptstyle 0}} c}$

Where $x{\scriptscriptstyle 0}, y{\scriptscriptstyle 0},z{\scriptscriptstyle 0}$ - point on that line

(a, b, c) - directing vector of a line

t is a parameter, $t \isin R$

(a, b, c) = (1 - 2, 4 - 1, -3 -3) = (-1, 3, -6)

Then

• vector form: $(x,y,z)=(2,1,3)+t(-1,3,-6)$
• parametric form: $x=2-t, y = 1+3t,z=3-6t$
• symmetric form: ${\frac {x-2} {-1}}={\frac {y-1} 3}={\frac {z-3} {-6}}$