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

Step 1: Identify the given point u005C(u005Cmathbf{S}(u002D5,1)u005C), the direction vector u005C(u005Cmathbf{v}u003D(4,u002D6)u005C

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.
$\mathbf{S}(-5,1) ; \mathbf{v}=(4,-6), \mathbf{T}(0,-1)$

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Key Concepts

Point-Line Distance

The distance from a point to a line is the shortest distance between the point and any point on the line, which is achieved along the perpendicular from the point to the line. This concept is a key application of analytic geometry and is essential for solving problems that involve relative positions of points and lines.

Vector Representation of a Line

A line can be expressed in vector form by specifying a point through which it passes and a direction vector that indicates its orientation. This representation makes it easier to manipulate lines, perform calculations like finding distances, and understand geometric relationships in a coordinate system.

Cross Product Method for Distance Calculation

Using the cross product is a robust method for calculating the distance from a point to a line when the line is given in vector form. By forming a vector from the line’s point to the given point and computing its cross product with the line’s direction vector, one obtains a vector whose magnitude is proportional to the area of the parallelogram formed. Dividing this magnitude by the magnitude of the direction vector yields the perpendicular distance from the point to the line.

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