# 12.5.4 Find parametric equations .

• MHB
• karush
In summary, to find the parametric equations of a line through a given point and perpendicular to two given vectors, you can use the vector product or solve a system of equations. The resulting equations will involve the given point and a parameter. It is also possible to multiply all coordinates of the vector product by a nonzero number to obtain equivalent equations. Care should be taken to avoid typos and double check the answers, as they may contain errors.
karush
Gold Member
MHB
$\textsf{Find parametric equations .}$
$\textsf{of the line through the point }$
$$P(-3, -4, -2)$$
$\textsf{and perpendicular to the vectors }$
$$u = -6i + 2j + 8k$$and $$v = -7i + 5 j - 2k$$
$\textit{Answer:$\displaystyle x = -44t - 3 , y = -68t - 4, z = -16t - 2 $}$
ok how is this done with 2 vectors

You can find a vector perpendicular to $(-6,2,8)$ and $(-7,5,-2)$ by computing the vector product of these vectors or by finding any nontrivial solution to the system of equations \displaystyle \left\{\begin{align}-6x+2y+8z&=0\\-7x+5y-2z&=0\end{align}\right..

$\displaystyle \begin{bmatrix} i & j & k\\ -6 &2 &8\\ -7 &5 &-2 \end{bmatrix} = \begin{bmatrix} 2&8\\ 5&-2 \end{bmatrix}i - \begin{bmatrix}-6&8\\-7&-2 \end{bmatrix}j + \begin{bmatrix}-6&2\\-7&5 \end{bmatrix}k =-44i-68j-16k\\$
$\text{so the parametric would be from P(-3,-4,-2)}$
\begin{align*}
x&=-44t-3,y=-68t-4,z=-16t-2
\end{align*}

Last edited:
The sign before the determinant multiplied by $j$ should be a minus. The equation for $x$ should be $x=-44t-3$ and for $z$ it should be $z=-16t-2$. You can multiply all coordinates of the vector (cross) product by any nonzero number, for example, by $-1/4$.

Evgeny.Makarov said:
The sign before the determinant multiplied by $j$ should be a minus. The equation for $x$ should be $x=-44t-3$ and for $z$ it should be $z=-16t-2$. You can multiply all coordinates of the vector (cross) product by any nonzero number, for example, by $-1/4$.

Well one thing nice here at MHB is the typos are shown
other forums rarely do that

## What is the purpose of finding parametric equations?

Finding parametric equations allows us to represent a curve or surface in terms of one or more independent parameters. This can provide a more concise and geometrically intuitive representation of a mathematical object.

## What is the process for finding parametric equations?

The process for finding parametric equations involves expressing the coordinates of a point on the curve or surface in terms of a parameter, such as t or u. These equations can then be used to plot the curve or surface and analyze its properties.

## What is the difference between parametric equations and Cartesian equations?

Parametric equations use parameters to describe the coordinates of points on a curve or surface, while Cartesian equations use variables to describe the relationship between x and y coordinates. Parametric equations can provide a more flexible and visual representation, while Cartesian equations are often simpler to work with algebraically.

## How do parametric equations relate to vector equations?

Parametric equations can be used to describe the path of a vector in space, as the coordinates of the vector can be expressed in terms of a parameter. This allows for a more dynamic and intuitive representation of vector motion.

## What are some common applications of parametric equations?

Parametric equations are commonly used in fields such as physics, engineering, and computer graphics to represent and analyze curves and surfaces. They can also be used to model real-world phenomena, such as motion, airflow, and population growth.

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