Two parameterized lines are given. Are they the same line? r⃗1(t)=(5-3 t) i⃗+(1+t) j⃗+2 t k

step 1 So we have two parameterized lines here. We want to see whether or not they are secretly the sam

Two parameterized lines are given. Are they the same line? ...

Two parameterized lines are given. Are they the same line?
$$\begin{array}{l}
\vec{r}_{1}(t)=(5-3 t) \vec{i}+(1+t) \vec{j}+2 t \vec{k} \\
\vec{r}_{2}(t)=(2+6 t) \vec{i}+(2-2 t) \vec{j}+(3-4 t) \vec{k}
\end{array}$$

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Key Concepts

Parameterized Line Equation

A parameterized line in vector form expresses each point on the line as the sum of a fixed position vector and a scalar multiple of a direction vector. This form is useful for describing lines in any dimensional space and facilitates the process of comparing two lines for similarity, parallelism, intersection, or coincidence by equating corresponding vector components.

Direction Vector

The direction vector in a parameterized line represents the orientation and slope of the line. It indicates the line’s direction and is critical when determining whether two lines are parallel or coinciding, as two identical lines should have proportional direction vectors along with overlapping points.

Line Coincidence

Determining if two lines are the same involves checking if their directional vectors are scalar multiples of one another and if one line’s point can be found on the other line. By equating the vector equations and considering parameter transformations, one can conclude whether the lines are identical (coincident) or simply parallel or intersecting in some other way.

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