The Equation of a Line in 3-Space

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In summary, to find the parametric equations of a line that intersect both L1 and L2 at right angles, we can use the slope vector method by identifying the parallel vectors for L1 and L2 and finding their cross product to determine the slope vector of the desired line. Then, we can use this slope vector to ensure that the desired line intersects both L1 and L2.
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Find parametric equations of a line that intersect both L1 and L2 at right angles.
L1: [x,y,z]=[4,8,-1] + t[2,3,-4] L2: (x-7)/-6 = (y-2)/1 = (z+1)/2
 
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Stephen10523 said:
Find parametric equations of a line that intersect both L1 and L2 at right angles.
L1: [x,y,z]=[4,8,-1] + t[2,3,-4] L2: (x-7)/-6 = (y-2)/1 = (z+1)/2

Each line is parallel to a vector -- can you identity a vector to which L1 is parallel? and likewise for L2? (Hint: think about the "slope" vector).

Once you have two vectors, it should be easy to find a vector perpendicular to these two (cross product). Think of this new vector as the "slope" vector of the desired line. All that is left is to ensure that the desired line intersects L1 and L2.
 

What is the equation of a line in 3-space?

The equation of a line in 3-space is a mathematical representation of a straight line in three dimensions. It is written in the form ax + by + cz = d, where a, b, and c are the coefficients of the x, y, and z variables, and d is a constant.

What is the significance of the coefficients in the equation of a line in 3-space?

The coefficients in the equation of a line in 3-space determine the slope and intercept of the line. The coefficient a represents the slope of the line in the x-direction, b represents the slope in the y-direction, and c represents the slope in the z-direction. The constant d determines where the line intersects the three axes.

How is the equation of a line in 3-space different from a 2-dimensional line?

The equation of a line in 3-space includes an additional variable, z, which represents the third dimension. This means that the line is not confined to a single plane, but rather extends into 3-dimensional space. Additionally, the coefficients in the equation are now represented by three variables instead of two, making it a more complex equation.

Can the equation of a line in 3-space be graphed?

Yes, the equation of a line in 3-space can be graphed on a 3-dimensional coordinate system. The x, y, and z axes represent the three dimensions, and the equation of the line can be plotted as a straight line through these three axes. This graph can provide a visual representation of the line's slope, intercepts, and direction in 3-dimensional space.

What are some real-world applications of the equation of a line in 3-space?

The equation of a line in 3-space is used in various fields such as engineering, physics, and computer graphics. It can be used to model the movement of an object through 3-dimensional space, calculate the shortest distance between two points in 3D, and create 3D visualizations of objects and structures. It is also commonly used in computer programming for tasks such as collision detection and path planning.

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