# S6.802.12.5.5 Find a vector equation and parametric equations

• MHB
• karush
In summary, we can find the vector equation and parametric equations for the line through the point (1,0,6) and perpendicular to the plane $x+3y+z=5$ by multiplying the normal vector to the plane, (1, 3, 1), by a parameter (t) and positioning it to go through the point, giving x = t + 1, y = 3t, and z = t + 6.
karush
Gold Member
MHB
see post 3

Last edited:
karush said:
$\tiny{s6.802.12.5.5}$
$\textsf{ Find a vector equation and parametric equations for:}\\$
$\textsf{The line through the point$(1,0,6)$}\\$
$\textsf{and perpendicular to the plane$x=-1+2t, y=6-3 t, z=3+9t$}$

$\textit{looking at some examples but? }$

What you have posted is not a plane, but a line. There are an infinite number of planes that that line could go through...

$\textsf{sorry copied problem incorrectly it should read}\\$

$\textsf{Find a vector equation and parametric equations for the line.}\\$

$\textsf{through the point$(1,0.6)$and perpendicular to the plane$x+3y+z=5$!.}$

$\textit{the book answer to this was}\\$
$r=(i+6k)+t(i+3j+k)\\$
$x=1+t, y=3t, z=6+t$
$\textit{but don't know how it was derived!}$

Last edited:
karush said:
$\textsf{sorry copied problem incorrectly it should read}\\$

$\textsf{Find a vector equation and parametric equations for the line.}\\$

$\textsf{through the point$(1,0.6)$and perpendicular to the plane$x+3y+z=5$!.}$

$\textit{the book answer to this was}\\$
$r=(i+6k)+t(i+3j+k)\\$
$x=1+t, y=3t, z=6+t$
$\textit{but don't know how it was derived!}$

A plane's normal vector always has the same coefficients as the plane, so the normal vector to the plane is (1, 3, 1). To make it infinitely long, multiply by a parameter (t) which can take on any real number, so t(1, 3, 1) = (t, 3t, t). Then position it so it can go through the point (1, 0, 6), giving (t + 1, 3t + 0, t + 6). Thus x = t + 1, y = 3t and z = t + 6.

## 1. What is a vector equation?

A vector equation is an equation that represents a relationship between two or more vectors. It typically includes variables, constants, and vector operations such as addition, subtraction, and scalar multiplication.

## 2. How is a vector equation different from a scalar equation?

A scalar equation involves only scalar quantities (numbers) and does not involve vectors or vector operations. A vector equation, on the other hand, involves vectors and vector operations and can represent more complex relationships between these quantities.

## 3. What are parametric equations?

Parametric equations are a set of equations that express the coordinates of a point in terms of one or more parameters. These equations allow for a more flexible and intuitive representation of curves and surfaces in space.

## 4. How do you find a vector equation from a set of parametric equations?

To find a vector equation from parametric equations, you can simply express the coordinates of the point as a vector with the parameter as the variable. For example, if the parametric equations are x = t, y = 2t, z = 3t, the vector equation would be r = ti + 2tj + 3tk.

## 5. Why are vector and parametric equations useful in science?

Vector and parametric equations are useful in science because they allow for a more precise and geometric representation of physical phenomena. They are especially useful in fields such as physics, engineering, and computer graphics, where calculations involving vectors and curves are common.

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