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I'm currently working on an exercise in analytical geometry in three-dimensional space, and I have a set of questions regarding the problem. The exercise involves two lines, $L1: {x = 2 − t, y = 3t − 1, z = 5t − 6}$ and $L2: {x = 5 + 2t, y = −t, z = 1 + t}$, along with several planes. My goal is to determine the relationship between these lines and planes. Here are my questions for each part of the problem:

Part (a): Parallel Lines

How can I determine whether two lines in three-dimensional space, such as $L_1$ and $L_2$, are parallel? What would be the appropriate approach for verifying the parallelism of these lines? Are there specific properties of the direction vectors of the lines that are crucial in making this determination?

Part (b): Perpendicular Lines

When dealing with two lines in three-dimensional space, how can I determine whether they are perpendicular to each other, as in the case of $L_1$ and $L_2$? What would be the correct approach for checking the perpendicularity? How should I calculate the dot product between the direction vectors of the lines to make this decision?

Part (c): Planes Containing Lines

What is the procedure for determining if a given plane, such as $8x + 11y - 5z = 35$, contains both of the specified lines? How can I verify whether a given plane contains two specific lines? What steps should I take to address this part of the problem effectively?

Part (d): Line and Plane Parallelism

How can I establish whether a line, $L_2: {x = 5 + 2t, y = −t, z = 1 + t}$, is parallel to a given plane, $2x - y + z = 0$, in three-dimensional space? What would be the correct strategy for verifying the parallelism between a line and a plane? How should I compare the normal vector of the plane with the direction vector of the line?

Part (e): Intersection of Lines at a Point

What is the proper procedure for determining if two lines, $L_1$ and $L_2$, intersect at the specific point $(1, 2, -1)$ in three-dimensional space? How can I determine if two lines intersect at a particular point? How should I solve the resulting system of equations to verify whether they meet at this point?

I appreciate your help and guidance on these questions, as I work towards solving this analytical geometry problem. Thank you in advance for your assistance!

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So lines are of the form $v_0+tv$ for some fixed vectors $v_0$ and $v$ and $t$ a real scalar. This line will be in the direction of $v$ and through $v_0$. So $L_1=[2,-1,6]^T + t[-1,3,5]^T$ and $L_2 = [5,0,1]^T+t[2,-1,1]^T$. To see if they're parallel we check if they're in the same direction. So if we write $L_1=v_0+tv$ and $L_2=u_0+tu$ then they will be parallel if there exists some non-zero scalar $s$ such that $sv=u$, which is to say that $v$ and $u$ are linearly dependent. To show that two lines intersect you have to find a $t_1$ and $t_2$ so that $v_0+t_1v=u_0+t_2u$. Determining if they're perpendicular is done by checking that the dot product of their direction vectors is zero.

Planes will be written in the form $P=n^Tv=[n_1,n_2,n_3]\cdot [x,y,z]=k$ with $k$ some constant. The vector $n$ is called the normal vector and is perpendicular to vectors in the plane. In your problem $n=[8,11,-5]^T$ and $k=35$. Here the plane and the line defined by the normal vector form orthogonal compliments of one another. This relationship shows up a lot and often problems involving planes can be rephrased in terms of the normal vector to simplify them. So to see if a line is parallel to a plane we want to check if lines direction is orthogonal to the normal vector of the plane.

To see if a line is in the plane substitute the $x,y$ and $z$ values into the planar equation and see if they're satisfied for all $t$. If they're only satisfied for one $t$ then that's the point of intersection. If they're never satisfied they're parallel which would provide a second method to detecting parallelism. All the specifics are linear equations and you can you use all the familiar techniques to solve them. Once you get used to the relationship between the normal vector and the plane the rest should fall into place.

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  • $\begingroup$ Thank you for your previous response. I have a question for you: In Part (c), I have the equation of the plane 8x + 11y - 5z = 35. When analyzing whether the line L1 is contained in this plane, I arrived at the equation 16 - 8t + 33t - 11 - 25t + 30 = 35, which simplifies to 0 = 0. Since this equation always holds, regardless of the value of t. Can we conclude that L1 is contained in this plane? Is the equation 0 = 0 sufficient to assert that a line is contained in a plane? I appreciate your help in clarifying this point. $\endgroup$
    – Ayesca
    Nov 1, 2023 at 4:38
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    $\begingroup$ @Ayesca exactly, the equation is always satisfied therefore it's in the plane. $\endgroup$ Nov 1, 2023 at 19:34

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