My question says:
Consider the vectors $a = i - j + k, b = i + 2j + 4k$ and $c = 2i - 5j - k.$
a) Given that $c = ma + nb$ where $m, n$ are real numbers, find the value of $m$ and $n$
b) Find a unit vector, $u$, normal to both $a$ and $b.$
For $(a)$ I have done this and got $n = -1$ and $m = 3$.
For $(b)$ I got $\frac{1}{\sqrt{118}}\left(-10i-3j+3k\right)$ as my unit vector.
But now I’m faced with
c) The plane $p_1$ contains the point $A (1, -1, 1)$ and is normal to $b.$ The plane intersects the x, y and z axes at the points $L, M$ and $N$ respectively:
i) Find the Cartesian equation of $p_1$
ii) Write doen the coordinates of $L, M$ and $N$
I’m not sure where to start at this point, can anyone help?
If the equation of plane is $ax+by+cz=d$ then its normal is $<a,b,c>$. So, $p_1$'s equation is $$1(x-1)+2(y+1)+4(z-1)=0$$ or $$x+2y+4z=3$$ which contains the point $A$. Converting it to intercept form: $$\frac x3+\frac y{\frac32}+\frac z{\frac34}=1$$ which gives $L=(3,0,0);M=\Bigl(0,\frac32,0\Bigr);N=\Bigl(0,0,\frac34\Bigr)$.