# Equation of the line given a translation vector (that is coincident) and a point

• singleton
In summary, the conversation discusses finding the equation of a line that passes through a given point and maps onto itself under a given translation vector. The line must have the same direction as the translation vector and the given point must lie on the line. Using this information, the equation of the line can be found in either slope-intercept form or standard form.

#### singleton

Hey,

Usually I have a general idea how to tackle a problem when posting on here.

This time I am pretty darn lost :(

When asked to find the equation of the line that passes through a point (1,1) and maps onto itself under the translation vector a = (3,2).

The equation of the image is
A(x - 3) + B(y - 2) + C = 0
Ax - 3A + By - 2B + C = 0

The point (1, 1) satisfies the equation since the translation "maps onto itself" (I'm taking this as it means a parallel and coincident line? not heard that terminology before..."maps onto itself") So the point is on both lines...

Basically I know that (1, 1) satisfies the equation of the image, and that when I substitute I end up with
Ax - 3A + By - 2B + C = 0
A(1) - 3A + B(1) - 2B + C = 0
-2A -B + C = 0

I almost wanted to say the normal vector for the translation was (-2, -1) but that is incorrect, as those numbers (-2 and -1) are the coefficients I have for A and B AFTER I substitute not BEFORE...

Just a small hint or suggestion how to tackle this Q would be great :)

I know (rather, I think) that I should be exploring the equation of the image (the translation) but I do not know what more I can do to find the normal vector (so I can write the Cartesian equation of the unknown line)

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If you translate the line by (3,2) then you get the same line back. For example, if you have the x-axis, and you translate it by (3,2), you get a different line, but if you translate it by (3,0) you get the same line. Translating a line will give you back the same line if and only if you translate in the direction that it's already "facing." So you can tell from this the direction vector of the line, and you already know one point that lies on the line, so you have all the necessary information to describe it (and give its Cartesian equation if you want).

It is my understanding that if the line "maps onto itself under a translation along vector a = (3,2)" we know that what this says is that the new line is the same line as before, correct?

Then, if we find the image of point (1,1) under the translation vector (3,2), which is point (4,3) we know that this point is on the line. We also know that the original point, (1, 1) is on the line since it is still the same line as before...

So the translation vector is also the same as the direction vector, (3,2) ?
And the vector equation is: r = (1,1) + t(3,2) ? (then find a normal vector and go from there)

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Yes to all questions.

thanks very much :)

Essentially, saying that the vector <3,2> translates the line into itself just means that the vector <3,2> points "in the same direction" as the line. Even more helpful: if (x,y) is a point on the line then so is (x+3, y+2).

Here you are told that the point (1,1) is on the line and the vector <3,2> translates the line into itself. (1,1) translated by <3,2> is (1+3,1+2)= (4, 3) so we know that the two points (1,1) and (3,2) are on the line.

Write the line y= mx+ b. (1,1) is on the line means 1= m(1)+ b or m+ b= 1. (3,2) is on the line meas 2= m(3)+ b or 3m+ b= 2. Subtracting m+ b= 1 from that gives
2m= 1 or m= 1/2. m+ b= 1 becomes 1/2+ b= 1 so b= 1- 1/2= 1/2.

The equation of the line is y= (1/2)x+ 1/2 or 2y= x+ 1.

You could, of course, use the form Ax+ By= C. Putting x= 1, y= 1 gives A+ B= C.
Putting x= 3, y= 2 gives 3A+ 2B= C. That's two equations for 3 unknowns but that is because multiplying Ax+ By= C by any number gives another equation for the same line. We can choose one of the numbers to be anything (except 0) that we like and still have an equation for the same line.

Subtracting A+ B= C from 3A+ 2B= C eliminates C leaving 2A+ B= 0 or B= -2A. Choosing A= 1 gives B= -2 and C= A+ B= 1- 2= -1.

The equation is x- 2y= -1 which is the same as 2y= x+ 1.

## 1. What is the equation of a line given a translation vector and a point?

The equation of a line can be expressed in the form y=mx+b, where m is the slope of the line and b is the y-intercept. To find the equation of a line given a translation vector and a point, you will need to use the point-slope form of the equation: y-y1=m(x-x1), where (x1, y1) is the given point and m is the slope calculated from the translation vector.

## 2. How do you determine the slope of a line using a translation vector?

The slope of a line can be determined using the formula y2-y1/x2-x1, where (x1, y1) and (x2, y2) are two points on the line. When given a translation vector, you can use the vector's components as the coordinates for your two points, and then plug them into the slope formula to find the slope of the line.

## 3. Can the equation of a line be determined with just a translation vector?

No, the equation of a line cannot be determined with just a translation vector. The translation vector only provides information about the direction and magnitude of the line's translation, but it does not give any information about the line's slope or y-intercept. To determine the equation of a line, you will need at least one point on the line in addition to the translation vector.

## 4. How does a translation vector affect the equation of a line?

A translation vector affects the equation of a line by changing the position of the line on the coordinate plane. The vector's components determine the amount and direction of the line's translation, which can then be used to calculate the slope and y-intercept of the line. These values are then used in the equation of a line to determine its position and slope.

## 5. Can a translation vector change the slope of a line?

Yes, a translation vector can change the slope of a line. The slope of a line is determined by the ratio of the change in y-coordinates to the change in x-coordinates between two points on the line. When a translation vector is applied to a line, it changes the position of the line on the coordinate plane, which in turn changes the coordinates of the points on the line. This change in coordinates can result in a different slope for the translated line.