Intersection of the two planes

Question

I need help with my vector's assignment! Let $L$ be the line of intersection of the two planes $x+y+z-1=0$ and $2x+3y-z+2=0$.

• Find the scalar equation of the plane that contains the line $L$ and passes through the origin.

Answer 1

The planes through the intersection line $L$ form a pencil of planes with equation $$\lambda(x+y+z-1)+\mu(2x+3y-z+2)=0$$ for a pair $(\lambda,\mu)$ defined up to a non-zero constant factor. If it passes through the origin, we obtain the condition $\;-\lambda+2\mu=0$, whence for instance $\mu=1,\lambda=2$.

Answer 2

As you are learning plane geometry i would consider that all your concepts on straight lines are clear and brushed

as in straight line you might have learned about family of straight lines similarly we can use family of planes passinf through line of intersection of two planes....for further information refer to your books

So the family of plane will have equation

{x+y+z}+A{2x+3y-z+2}=0 ..... where a is your non zero constant of proportionality

Rearranging the equation ... x{1+2A}+y{1+3A}+z{1-A}+{-1+2A}=0

According to the problem the plane also passes through the origin ie {0,0,0}

{0}{1+2A}+{0}{1+3A}+{0}{1-A}+{-1+2A}=0 as it satisfies the equation

we get 2A-1=0......A=1/2

Puting the values

{x+y+z}+{0.5}{2x+3y-z+2}=0