find the distance between the given point 𝐒 and the line having the given direction vector v or

Step 1: Identify the coordinates of point u005C(u005Cmathbf{S}u005C) and point u005C(u005Cmathbf{T}u005C). We have u005C(u005Cma

Find the distance between the given point 𝐒 and the line...

find the distance between the given point $\mathbf{S}$ and the line having the given direction vector $v$ or slope $m$ and passing through the given point T.
$\mathbf{S}(-3,7) ; m=-\frac{2}{3}, \mathbf{T}(-4,6)$

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Key Concepts

Distance from a Point to a Line

This concept involves determining the shortest distance (or the length of the perpendicular) from a given point to a line. The idea is based on the fact that the minimal distance is measured along a line that is perpendicular to the given line. This distance can be calculated using various formulas that incorporate the line’s slope or direction vector.

Line Representation in the Plane

A line in the two-dimensional plane can be defined in several ways, including the point-slope form, slope-intercept form, or vector form. Each of these representations uses different parameters such as a fixed point on the line and a direction vector or slope, and they are essential for understanding the geometric and algebraic properties of the line.

Perpendicular Projection

Perpendicular projection refers to projecting a vector—in this case, the vector connecting an external point to a point on the line—onto a line that is perpendicular to the given line. This method is fundamental in identifying the exact point on the line that is closest to the external point, thereby allowing the calculation of the shortest distance.

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