# Determining coordinates of a point on a line perpendicular to a vector

In summary: No problem! I'll check them out when I have the time.In summary, the homework statement is trying to determine the coordinates of a point B on L2 such that vector AB is perpendicular to L2.

## Homework Statement

The two lines L1: r = (-1,1,0)+ s(2,1,-1) and L2: r = (2,1,2) + t(2,1,-1) are parallel but do not coincide. The point A(5,4,-3) is on L1. Determine the coordinates of a point B on L2 such that vector AB is perpendicular to L2.

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## The Attempt at a Solution

I'm not sure where to start on this one. I know that eventually AB dot (2,1,-1) will equal 0, but I'm not really sure what to do with this question.

Welcome to PF!

… I know that eventually AB dot (2,1,-1) will equal 0 …

That's right!

So just write B in terms of t, and chug away.

How do you do that? That's my main source of confusion here. How to write B in terms of t?

Every point on L2 has to satisfy L2's equation, which means that every point on L2 has coordinates of (2 + 2t, 1 + t, 2 -t) for some value of t.

So I do (5,4,-3)-(2+2t,2+t,2-t) to get AB and then do AB dot (2,1,-1) and find t somehow?

Almost. AB = <2 + 2t, 1 + t, 2 - t> - <5, 4, -3>. The dot product of AB and <2, 1, -1> should be 0.

Great! Thanks! I got it! I have a sort of general question for you that I don't think is worthy of its own thread really. I was just wondering what makes a line parallel to a plane?

A line is parallel to a plane if it (the line) is parallel to some line segment in the plane.

Another way to say this is that the line is parallel to the plane if it is perpendicular to the plane's normal.

Thanks a lot! That helped with a big question! I'm starting a couple more threads with other questions if you want to look at them.

## 1. How do you determine the coordinates of a point on a line perpendicular to a vector?

To determine the coordinates of a point on a line perpendicular to a vector, you can use the dot product or cross product method. The dot product method involves finding the dot product between the given vector and the point to determine a scalar value, which can then be used to calculate the coordinates of the perpendicular point. The cross product method involves taking the cross product between the given vector and another vector that is perpendicular to it, which will result in a vector that is also perpendicular to the given vector. This resulting vector can then be used to calculate the coordinates of the perpendicular point.

## 2. What is the dot product method for determining coordinates of a point on a line perpendicular to a vector?

The dot product method involves finding the dot product between the given vector and the point to determine a scalar value, which can then be used to calculate the coordinates of the perpendicular point. This method is based on the fact that the dot product of two perpendicular vectors is equal to 0. By setting up an equation with the dot product, you can solve for the coordinates of the perpendicular point.

## 3. How does the cross product method work for determining coordinates of a point on a line perpendicular to a vector?

The cross product method involves taking the cross product between the given vector and another vector that is perpendicular to it, which will result in a vector that is also perpendicular to the given vector. This resulting vector can then be used to calculate the coordinates of the perpendicular point. This method is based on the fact that the cross product of two perpendicular vectors is equal to a vector that is perpendicular to both of the original vectors.

## 4. Can you explain the geometric concept behind determining coordinates of a point on a line perpendicular to a vector?

The geometric concept behind determining coordinates of a point on a line perpendicular to a vector is based on the idea that a vector can be broken down into two components - one that is parallel to the given vector, and one that is perpendicular to it. By finding the perpendicular component of the vector, you can determine the coordinates of the point on the line that is perpendicular to the vector.

## 5. Are there any other methods for determining coordinates of a point on a line perpendicular to a vector?

Yes, there are other methods for determining coordinates of a point on a line perpendicular to a vector, such as using the slope-intercept form of a line or using trigonometric functions. However, the dot product and cross product methods are the most commonly used methods in vector calculations.

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