a. Show that the lines L1: r⃗=(-2,3,4)+s(7,-2,2), s ∈𝐑, and L2: r⃗=(-30,11,-4)+t(7,-2,2), t ∈𝐑,

step 1 Here we are given a pair of lines L1 and L2, and we are asked to show that they are coincident b

A. Show that the lines L1: r⃗=(-2,3,4)+s(7,-2,2), s ∈𝐑, and...

a. Show that the lines $L_{1}: \vec{r}=(-2,3,4)+s(7,-2,2), s \in \mathbf{R},$ and $L_{2}: \vec{r}=(-30,11,-4)+t(7,-2,2), t \in \mathbf{R},$ are coincident by writing each line in parametric form and comparing components b. Show that the point (-2,3,4) lies on $L_{2}$. How does this show that the lines are coincident?

a. Show that the lines $L_{1}: \vec{r}=(-2,3,4)+s(7,-2,2), s \in \mathbf{R},$ and $L_{2}: \vec{r}=(-30,11,-4)+t(7,-2,2), t \in \mathbf{R},$ are coincident by writing each
line in parametric form and comparing components
b. Show that the point (-2,3,4) lies on $L_{2}$. How does this show that the lines are coincident?

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Key Concepts

Vector Equation of a Line

A line in space can be expressed by the vector equation r = r0 + tv, where r0 is a fixed position vector (representing a point on the line) and v is a direction vector. This form is essential for representing lines in three dimensions and allows easy conversion to parametric equations.

Parametric Form of a Line

In the parametric form, each coordinate of a point on the line is written as a function of a single parameter (for example, x = x0 + ta, y = y0 + tb, and z = z0 + tc). This breakdown helps in comparing the components of different lines to analyze their relationships and intersections.

Parallel Lines and Coincidence

When two lines share the same direction vector, they are parallel by definition. However, if a point on one line is also on the other, then the lines are not just parallel but coincident, meaning they lie exactly on top of each other.

Point Inclusion in a Line

Verifying that a specific point lies on a given line involves substituting the point's coordinates into the line’s parametric equations to solve for the parameter. If a consistent solution is found, it confirms that the point is on the line, which, when coupled with a common direction vector, indicates the lines are coincident.

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