(a) Find a parametric equation for the line through the...

(a) Find a parametric equation for the line through the point (1,5,2) and in the direction of the vector $2 \vec{i}+3 \vec{j}-\vec{k}$ (b) By minimizing the square of the distance from a point on the line to the origin, find the exact point on the line which is closest to the origin.

(a) Find a parametric equation for the line through the point (1,5,2) and in the direction of the vector $2 \vec{i}+3 \vec{j}-\vec{k}$
(b) By minimizing the square of the distance from a point on the line to the origin, find the exact point on the line which is closest to the origin.

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Key Concepts

Parametric Equations of a Line in 3D

This concept involves describing a line in three-dimensional space using a point and a direction vector. The general form is given as r(t) = r0 + t*v, where r0 is a fixed position vector representing a point on the line, v is the direction vector, and t is a parameter. This representation allows every point on the line to be uniquely described by adjusting the parameter t.

Distance Minimization and Optimization

This concept deals with finding the point on a given line that is closest to a specific external point, in this case, the origin. Instead of working directly with the distance, which involves a square root, we optimize by minimizing the square of the distance function. By expressing the squared distance as a function of the parameter from the line's equation and then using calculus to find its minimum (usually by taking the derivative with respect to the parameter and setting it to zero), we determine the exact parameter value that yields the closest point.

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