Equations of lines Find both the parametric and the vector...

Equations of lines Find both the parametric and the vector equations of the following lines.
The line through (0,2,1) that is perpendicular to both $\mathbf{u}=4 \mathbf{i}+3 \mathbf{j}-5 \mathbf{k}$ and the $z$ -axis

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Key Concepts

Parametric Equations of a Line

Parametric equations express the coordinates of points on a line as functions of a parameter. Each coordinate (x, y, z) is defined as the sum of a fixed initial coordinate and a multiple of a corresponding component of the direction vector. This representation is particularly useful for describing lines in three-dimensional space, as it clearly separates the position and direction aspects of the line.

Vector Equation of a Line

The vector equation of a line represents all points on the line as the sum of a fixed position vector and a scalar multiple of a direction vector. This form, typically written as r = r0 + tv, succinctly describes the line by specifying a point on the line and its direction, simplifying calculations and geometrical reasoning in 3D space.

Direction Vectors and the Cross Product

A direction vector defines the orientation of a line. When a line is required to be perpendicular to two given directions, the cross product of the two vectors can be used to compute a new vector that is orthogonal to both. This new vector then serves as the direction vector for the line, ensuring that the line meets the perpendicularity conditions.

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