Vector Equation of Straight Line: y=4x+3

  • Thread starter blackenedrose
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Here, x= lambda and we are given y= 4x+ 3 so that x= lambda and y= 4 lambda+ 3. The vector equation is then r= lambda i+ (4 lambda+ 3)j.
  • #1
blackenedrose
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Homework Statement


find a vector equation for the straight line given by y=4x+3


Homework Equations


y=mx+c


The Attempt at a Solution



i think i need to change this into cartiesen then to parametrics but have no idea how to change that equation into cartisean
 
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  • #2
Welcome to PF rose,

The equation is already in Cartesian form, all you need to do now is parameterise it.
 
  • #3
Ok, thanks for pointing that out but as a person who has never been taught any of this before any chance of an example of how to do that? I've done it before when it has been for exampl r1=i+2j+2K+ landa(i-j+3k) yes i know how to get that into parametrics but not if its just y=4x+3
 
  • #4
Okay, you have done this when you have 3 dimensions and I think you are saying you can find the vector equation of a line through 2 given points: the line through (1, 2, 2) and (2, 1, 5) has "direction vector" (2-1)i+ (1-2)j+ (5-2)k= i- j+ 3k so we can write the vector equation as r= (i+ 2j+ 2k)+ lambda(i- j+ 3k).

One problem you have is that the single equation y= 4x+ 3 is that a single equation in 3 dimensions describes a plane, not a line. So you must mean "in the xy-plane" which means that z= 0.

Now, one way to do that is to find two points! if x= 0, y= 3 so one point the line passes through is (0, 3, 0). If x= 1, y= 7 so another point is (1, 7, 0). Use exactly the same method as above to find the vector equation of the line through those two points. You should see that the component of the k vector is 0: the line is always in the xy-plane, of course.

One thing that may be confusing you is that the is no one vector equation. The parameter, lambda, has no "geometrical" meaning and can be chosen almost arbitrarily. When we have all the other coordinates written as functions of the other (here y is a function of x) we can always use that one coordinate as the parameter. That is, if y= f(x), we can write x= lambda, y= f(lambda) and have the vector equation r= lambda i+ f(lambda) j.
 

Related to Vector Equation of Straight Line: y=4x+3

1. What is a vector equation of a straight line?

A vector equation of a straight line is a mathematical expression that represents a line in two or three dimensional space using vector notation. It is written in the form of r = a + tb, where r is the position vector, a is a fixed vector on the line, and b is a direction vector.

2. How is the vector equation of a straight line different from the standard form of a linear equation?

The vector equation of a straight line is different from the standard form of a linear equation in that it uses vector notation and allows for representation of lines in multiple dimensions. The standard form of a linear equation is written as y = mx + b, where m is the slope and b is the y-intercept.

3. What does the vector equation y=4x+3 represent visually?

The vector equation y=4x+3 represents a line with a slope of 4 and a y-intercept of 3. This line is a straight line that passes through the point (0,3) and has a slope of 4, meaning the line rises 4 units for every 1 unit it runs to the right.

4. How can the vector equation be used to find the coordinates of points on the line?

The vector equation can be used to find the coordinates of points on the line by plugging in different values for t. As t increases or decreases, the position vector r will change, giving the coordinates of different points on the line.

5. Can the vector equation be used to find the distance between two points on the line?

Yes, the vector equation can be used to find the distance between two points on the line. The distance between two points (x1,y1) and (x2,y2) on the line is given by the formula d = sqrt((x2-x1)^2 + (y2-y1)^2).

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