# Vector Perpendicular: Finding Point P from A & B

• thomas49th
In summary, the equation for the line through points A and B is i - 5j - 7k + t(9i + 15j + 12k). To find the position vector of point P such that OP is perpendicular to AB, set AB.OP = 0 and solve for t. Then, use the value of t to find the coordinates of P on the line.
thomas49th

## Homework Statement

The equation for the line that goes through vector points A and B is

i - 5j - 7k + t(9i + 15j + 12k)

A is defined: i - 5j - 7k
B is defined: 10i + 10j + 5k

Find the position vector of point P such that OP is perpendicular to AB

## The Attempt at a Solution

well a.b = 0 to be perpendicular

but how does that help...

Im lost

Any ideas. My test is at 9am tomorrow (GMT). Can someone just explain the whole method for me.

Thanks

(9i + 15j + 12k) is your direction vector

so you need <a,b,c> direction vector such that
<a,b,c> . (9, 15, 12,) = 0

pick any values for a,b,c

so the line perpendicular is r = (0,0,0) + t <a,b,c>

now, find intersection

thomas49th said:
The equation for the line that goes through vector points A and B is

i - 5j - 7k + t(9i + 15j + 12k)

A is defined: i - 5j - 7k
B is defined: 10i + 10j + 5k

Find the position vector of point P such that OP is perpendicular to AB

well a.b = 0 to be perpendicular

Hi thomas49th!

(btw, B isn't 10i + 10j + 5k, is it? )

I assume P is to be on AB?

ok, then OP must be i - 5j - 7k + t(9i + 15j + 12k)
for some value of t.

And, as you say, AB.OP = 0.

So … ?

I got to go to bed now … :zzz:

## 1. How do you find the point P that is perpendicular to points A and B?

To find the point P that is perpendicular to points A and B, first determine the slope of the line passing through points A and B. Then, take the negative reciprocal of that slope to find the slope of the perpendicular line. Use this new slope and the coordinates of points A and B to write an equation for the perpendicular line. Finally, solve this equation for the x and y coordinates of point P.

## 2. Can you explain the concept of vector perpendicularity?

Vector perpendicularity refers to the relationship between two lines that intersect at a right angle, forming a 90 degree angle. This means that the slopes of these lines are negative reciprocals of each other. In other words, when one line has a slope of m, the perpendicular line has a slope of -1/m.

## 3. How does vector perpendicularity relate to geometry and physics?

In geometry, vector perpendicularity is used to determine the relationship between two lines or planes. It is also a fundamental concept in understanding geometric shapes, such as triangles and quadrilaterals. In physics, vector perpendicularity is used to calculate forces and motion in three-dimensional space.

## 4. Can you find the point P that is perpendicular to a line segment, rather than two points?

Yes, you can find the point P that is perpendicular to a line segment by first finding the midpoint of the line segment. Then, using the same method as described in question 1, find the equation of the perpendicular line passing through the midpoint. This will give you the coordinates of the point P.

## 5. Are there any real-world applications of vector perpendicularity?

Yes, vector perpendicularity has numerous real-world applications. It is used in architecture and engineering to ensure that buildings and structures are stable and strong. It is also used in navigation and mapping to determine the direction of magnetic north. In addition, vector perpendicularity is used in computer graphics to create 3D images and animations.

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