# Points, Vectors, and Planes Oh My! (Finding parallel vectors and intersecting vectors)

I'm quite stumped on the following problem: Consider the planes 4x + 1y + 1z = 1 and 4x + 1z = 0. (A) Find the unique point P on the y-axis which is on both planes. (_, _, __)

(B) Find a unit vector u with positive first coordinate that is parallel to both planes __I + __J + _K

(C) Use the vectors found in parts (A) and (B) to find a vector equation for the line of intersection of the two planes r(t) = __I + __J + _K

Work thus far: I've figured out (A) is (0,1,0) Now, I know that the dot product of <4,1,1> and u will = 0, as well as the dot product between <4,0,1> and u... I'm quite stumped. Can anyone help me out by giving me hints? I'd prefer that the entire solution isn't given yet, so I can work through it. I'll respond as quickly as I can.

Thank you,

Landon

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(B) Write $\mathbf{u}=(a,b,c)$. You want $(4,1,1)\cdot (a,b,c) = 0$, so that means $4a+b+c=0$. You also want $(4,0,1)\cdot(a,b,c)=0$< so that means $4a+c=0$. This tells you that you must have $b=0$, and $c=-4a$. So the general form of vectors that are parallel to both planes is $(a,0,-4a)$. Since you want it to be a unit vector, you want $a^2 + 0^2 + (-4a)^2 = 1$, or $17a^2=1$. This gives you the absolute value of $a$ by solving for $a$, from which you can find one with positive first coordinate.