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

(a) Find a parametric equation for the line through the point (2,1,3) and in the direction of $a \vec{i}+b \vec{j}+c \vec{k}$.
(b) Find conditions on $a, b, c$ so that the line you found in part (a) goes through the origin. Give a reason for your answer.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Parametric Equations of a Line

This concept involves representing a line in space using one or more parameters. Typically, a line is expressed as r(t) = r? + tv, where r? is a fixed point on the line, v is the direction vector, and t is a scalar parameter that varies over all real numbers. This formulation allows one to easily describe every point on the line as a function of t.

Direction Vector

The direction vector describes the direction in which the line extends in space. It provides the rates at which the coordinates change as the parameter varies and is essential in defining the orientation of the line. The components of this vector indicate how much the x, y, and z coordinates increase or decrease when moving along the line.

Consistency Conditions for Intersection with a Specific Point

When determining if a given point lies on a parametric line, one must find a common parameter value that satisfies all the coordinate equations simultaneously. This involves setting the parametric expressions equal to the corresponding coordinates of the specific point and ensuring that the resulting parameter value is consistent across all coordinates, leading to conditions on the parameters or the components of the direction vector.

Transcript

Please give Ace some feedback