Create a vector-valued function whose graph matches the...

Create a vector-valued function whose graph matches the given description.
The line through points (1,2,3) and $(4,5,6),$ where $\vec{r}(0)=\langle 1,2,3\rangle$ and $\vec{r}(1)=\langle 4,5,6\rangle$

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Key Concepts

Linear Interpolation

Linear interpolation is used to find intermediate points on a line segment between two known points. By weighting these points with a parameter that varies between 0 and 1, one can seamlessly transition from one endpoint to the other, which is a common technique in parametrizing lines.

Position Vectors in 3D

Position vectors in three-dimensional space represent points by their coordinates. They form the basis for describing locations in space and are essential for constructing vector equations of lines, planes, and other geometric figures.

Vector-Valued Functions

A vector-valued function is a function that takes one or more parameters and returns a vector, typically representing points in space. This concept is essential in multivariable calculus and physics, as it describes curves and other geometric objects in a coordinate system.

Line Parametrization

Line parametrization involves expressing a line in space as a function of a parameter, typically using a point on the line and a direction vector. This method allows one to easily compute any point along the line by adjusting the parameter value.

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