If it exists, find the value of c for which the lines l(t)= (c+t, 1+t, 5+t) and m(s)=(s, 1-s, 3+

step 1 So we want these two lines to intersect, right? Let's write them out in this semi -expanded form

If it exists, find the value of c for which the lines l(t)=...

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Key Concepts

Parametric Equations

Parametric equations represent lines by expressing each coordinate as a function of one or more parameters. In three-dimensional space, a line is usually written in the form (x, y, z) = (a, b, c) + t(d, e, f), where (a, b, c) is a point on the line and (d, e, f) is the direction vector. This form allows easy manipulation when determining relations such as intersections.

Intersection of Lines

When two lines in space intersect, there exist specific values for their respective parameters that result in the same point in space. To determine intersection, one sets the parametric equations of the lines equal to each other component-wise, ensuring that the x, y, and z coordinates match simultaneously.

Systems of Linear Equations

Equating the coordinates of the parametric equations from two lines leads to a system of linear equations. Solving this system determines the parameter values and any conditions on constants that must hold for the lines to intersect. Consistency of these equations is crucial, as it indicates the existence of a common solution, i.e., an intersection point.

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