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}(4, u002D7)u005C), the direction vector u005C(u005Cmathbf{v} u003D (u002D5,

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.
$\mathbf{S}(4,-7) ; \mathbf{v}=(-5,-6), \mathbf{T}(1,0)$

Key Concepts

Determinant/Cross Product Method in 2D

In two dimensions, the distance from a point to a line can also be computed using the magnitude of the determinant (analogous to the cross product in three dimensions) of two vectors: one representing the displacement from a point on the line to the external point and the other the line’s direction vector. Dividing this magnitude by the norm of the direction vector yields the perpendicular distance.

Orthogonal Projection and Perpendicularity

Finding the distance from a point to a line involves projecting a vector (from a point on the line to the external point) onto a direction perpendicular to the line. The orthogonal projection method is used to identify the component of the vector that is perpendicular to the line, which directly gives the shortest distance.

Vector Representation of a Line

A line in the plane can be represented using a point on the line and a direction vector. The direction vector specifies the orientation of the line, while the fixed point anchors the line in space. This representation is fundamental in vector geometry since it facilitates the use of vector operations to analyze geometric relationships, such as computing distances and intersections.

Distance from a Point to a Line

This concept involves finding the shortest distance (perpendicular distance) from a point to a line. In analytic geometry and vector analysis, this is often calculated using a formula that involves the magnitude of the vector difference between the point and any point on the line, together with a component perpendicular to the line. This approach ensures that the shortest route—that is, the perpendicular route—is measured.

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