Vector equation of line with angle to a plane

**nvwxgn** · June 28th 2010, 01:02 PM

Problem is: Find the vector equation of the line r that contains the point P (1, 1, 1), is parallel to the plane x + 2y - z = 0 and presents a pi/3 radians angle with the plane x - y + 2z - 1 = 0

I let the direction vector of r be (a, b, c) and consider that (a, b, c).(1, 2, -1) = 0, since it's parallel to the first plane. Also, sqrt(3)/2 = (a, b, c).(1, -1, 2) / |a, b, c| times sqrt(6). But then, I'm stuck at a three-variable system with two equations and also a modulus.

How am I supposed to do that? Thanks in advance.

**gmatt** · June 28th 2010, 03:01 PM

Consider the information that you are given carefully, I will help.

The line $r$ is parallel to the plane $P_1$ defined by $\vec{x}\cdot (1,2,-1) = 0.$ This means that it lies in $P_1$ and as you have correctly deduced, if we let the direction vector of $r$ be $\vec{a}=(a,b,c)$ then we require that $\vec{a} \cdot (1,2,-1) = 0$ . Let $P_2$ be the plane defined by $\vec{x} \cdot(1,-1,2) = 1$ .

Now consider the intersection of $P_1$ and $P_2$ , $P_1 \cap P_2.$ Clearly $P_1 \cap P_2$ is a line with direction vector $\vec{b}=(1,2,-1) \times (1,-1,2)$ , and furthermore $P_1 \cap P_2$ is parallel with both $P_1$ and $P_2$ . For $r$ to have $\pi / 3$ angle with $P_2$ we must have that $r$ and $P_1 \cap P_2$ have angle $\pi / 3$ since $r$ is parallel to $P_1$ , and $P_1 \cap P_2$ is parallel with both $P_1$ and $P_2$ .

Thus, we require that $\frac{\vec{a} \cdot \vec{b}}{\parallel \vec{a} \parallel \parallel \vec{b} \parallel} = \cos ( \pi / 3 ).$

You have two conditions so far:

1) $\vec{a} \cdot (1,2,-1) = 0$
2) $\frac{\vec{a} \cdot \vec{b}}{\parallel \vec{a} \parallel \parallel \vec{b} \parallel} = \cos ( \pi / 3 )$ .

The final (third) condition on $\vec{a}$ is rather artificial. Lets put the condition on $\vec{a}$ so that we find a unit vector, that is the 3 conditions are then

1) $\vec{a} \cdot (1,2,-1) = 0$
2) $\frac{\vec{a} \cdot \vec{b}}{ \parallel \vec{b} \parallel} = \cos ( \pi / 3 )$ .
3) $\left\parallel \vec{a} \right\parallel = 1.$

I think that what I wrote above is correct; however, I am not sure why you are given the point through which the line $r$ must pass through, I believe its a red herring.

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