Solving for a Cartesian Equation of a Line with Given Vector and Point

In summary, to find the Cartesian equation for the line p + tv, where p=<2,5> and v=<-3,1>, you can use the vector form <2,5>+t<-3,1> and substitute two points into the line y=mx+b to find m and b. Alternatively, you can eliminate t from the system x=2-3t and y=5+t and solve for one variable to then substitute into the other equation.
  • #1
Jeann25
30
0

Homework Statement



Find a Cartesian equation for the line p + tv, where p=<2,5> and v=<-3,1>

Homework Equations





The Attempt at a Solution



I know the vector form will look likse <2,5>+t<-3,1>. But I'm not sure how to use this information to find the equation of the line in y=mx+b form.
 
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  • #2
Just substitute two points into the line y=mx+b to find m and b. Also, the slope m can be obtained immediately from the direction vector v.
 
  • #3
Or, if you write the vector form in components, eliminate t from this system:

[tex]
\left\{ \begin{array}{l}
x = 2 - 3t \\
y = 5 + t \\
\end{array} \right.
[/tex]
 
  • #4
Now, solve one equation for t and (I recomment y= 5+ t) and substitute that into the other equation.
 

What is a Cartesian equation of a line?

A Cartesian equation of a line is an algebraic equation that represents a straight line on a Cartesian plane. It is in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

How do you solve for a Cartesian equation of a line with a given vector and point?

To solve for a Cartesian equation of a line with a given vector and point, you can use the point-slope form of a line. Plug in the coordinates of the given point and the components of the vector into the formula y - y1 = m(x - x1). Then, simplify for y to get the Cartesian equation of the line.

What is the point-slope form of a line?

The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. This form is useful for finding the equation of a line with a given point and slope.

Can a Cartesian equation of a line have a negative slope?

Yes, a Cartesian equation of a line can have a negative slope. This means that the line is decreasing from left to right on the Cartesian plane.

What is the relationship between a vector and a Cartesian equation of a line?

A vector represents the direction and magnitude of a line. It can be used to find the slope of the line, which is an important component of the Cartesian equation of a line. Without a vector, it would be difficult to determine the equation of a line on a Cartesian plane.

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