Multivariable calculus and vector analysis

Parametrization of a line examples

Example 1

Find a parametrization of the line through the points $\left(3,1,2\right)$ and $\left(1,0,5\right)$.

Solution: The line is parallel to the vector $\mathbf{v}=\left(3,1,2\right)-\left(1,0,5\right)=\left(2,1,-3\right)$. Hence, a parametrization for the line is

$$\begin{array}{lll}\hfill \mathbf{x}=\left(1,0,5\right)+t\left(2,1,-3\right)\text{ for}-\infty <t<\infty .& & \hfill \end{array}$$

We could also write this as

$$\begin{array}{lll}\hfill \mathbf{x}=\left(1+2t,t,5-3t\right)\text{ for}-\infty <t<\infty .& & \hfill \end{array}$$

Or, if we write $\mathbf{x}=\left(x,y,z\right)$, we could write the parametric equation in component form as

$$\begin{array}{llll}\hfill x& =1+2t,& \hfill & \\ \hfill y& =t,& \hfill & \\ \hfill z& =5-3t,& \hfill & \end{array}$$

for $-\infty <t<\infty$.

Example 2

Find a parametrization for the line segment between the points $\left(3,1,2\right)$ and $\left(1,0,5\right)$.

Solution: The only difference from example 1 is that we need to restrict the range of $t$ so that the line segment starts and ends at the given points. We can parametrize the line segment by

$$\begin{array}{lll}\hfill \mathbf{x}=\left(1,0,5\right)+t\left(2,1,-3\right)\text{ for}0\le t\le 1.& & \hfill \end{array}$$

(Or we could use any of the other forms of the parametric equation in example 1, as long as we restrict $t$ to lie in the interval $\left[0,1\right]$.)

Contents by topic

1. Geometry in higher dimensions

1a. Preliminaries

1b. Lines and planes

• Parametrization of a line
• Forming planes
• A line or a plane or a point?
• Parametrization of a plane
• Distance from point to plane
• Parametrization of a line examples
• Forming plane examples
• Plane parametrization example
• Distance from point to plane example
• Intersecting planes example

1c. Matrices

1d. Visualizing functions in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

5. The fundamental theorems of vector calculus

6. Review material