# Area of a Triangle Using Vectors.

1. May 25, 2014

### smakhtar

1. The problem statement, all variables and given/known data
A triangle has verticies A(-2,1,3), B(7,8,-4), and C(5,0,2). Determine the area of the triange ABC.
The correct answer is 35.9 square units.

2. Relevant equations
Has to be done by using dot product and/or cross product.

Dot product: a(dot) b= |a||b|cos(theta)
Cross product: a x b= |a||b|sin(theta)
||- this is used to indicate magnitude

3. The attempt at a solution
AC=(5-(-2),0-1,2-3)
=(7,-1,1)
AB= (7-(-2),8-1,-4-3)
=(9,7,-7)
BC=(5-7,0-8,-1-4)
=(-2,-8,6)

I tried using this formula AB (dot) BC x AC, and I got 358 units squared, which is incorrect. I am trying to get to the correct answer mentioned above. What would be the formula that would give me the correct answer? Please reply as soon as possible

2. May 25, 2014

### SammyS

Staff Emeritus
The scalar triple product, AB (dot) BC x AC, gives a volume, so it's in cubed units.

What does CB x CA represent geometrically?

3. May 25, 2014

### smakhtar

It has to be done using Cartesian vectors not geometric.

4. May 25, 2014

### SammyS

Staff Emeritus
But the point here is to help you understand how to answer the question, eventually using vectors. And the question asked does refer to vectors.

To repeat:

What does the cross-product (vector product), CB × CA, represent geometrically?

5. May 25, 2014

### smakhtar

CA and CB are vectors. It geometrically represents the right hand rule where the thumb represents the z axis.

6. May 25, 2014

### smakhtar

And the finders and palm would refer to the x and y axis.

7. May 25, 2014

### SammyS

Staff Emeritus
What does the magnitude of CB × CA represent ?

8. May 25, 2014

### smakhtar

The length of the sides.

9. May 25, 2014

### SammyS

Staff Emeritus
No.

The magnitude of each vector is the length of each side.

What's the magnitude of the cross-product?

10. May 25, 2014

### smakhtar

The magnitude of cross product equals the magnitude of |CB||CA|sin (theta).

11. May 25, 2014

### Nathanael

Yes, but there's a visual meaning of the cross product as well.

Perhaps you didn't learn (or don't remember) it so I'll just tell you, CA X CB is equal to the area of the parallelogram with sides being |CA| and |CB| and the angle between the sides is the angle between the CA and CB (what you called theta)
(From the definition of "parallelogram," the other two sides are parallel to CA and CB.)

So what would the area of triangle with points (A,B) (A,C) and (B,C) be?

12. May 25, 2014

### smakhtar

I am assuming that you are referring to the coordinates mentioned above. I think it would be (AB x BC) (Dot) CA.

13. May 25, 2014

### Nathanael

That would be a volume. Draw the triangle and draw the parallelogram I described (AB X BC) and see if you can find a relationship.

14. May 25, 2014

### SammyS

Staff Emeritus
Use the information Nathaniel gave you.

I already indicated that a scalar triple product, such as AB · (BC × AC) is a volume. By the way, your result for that was incorrect in your Original Post. For any three points A, B, C, that triple product is a vector of zero magnitude.

By the way, you could consider |CA|sin (θ) to be the altitude of a triangle and |CB| as the base.

15. May 25, 2014

### smakhtar

So it would be A= |CB|(|CA|sin (theta) all over 2?

16. May 25, 2014

### SammyS

Staff Emeritus
(Don't use ' A ' for multiple purposes -- a point and an area .)

Yes that will work, but consider the following.

|CB|(|CA|sin (θ) is the magnitude of the cross-product of vectors CB and CA .

and

You can use any two of the three following vectors that you previously determined.
AC=(5-(-2),0-1,2-3)
=(7,-1,1)
AB= (7-(-2),8-1,-4-3)
=(9,7,-7)
BC=(5-7,0-8,-1-4)
=(-2,-8,6)​

Finally, CB x CA is a vector perpendicular to the plane in which points A, B, and C lie, having the magnitude of the parallelogram with adjacent sides formed by CB and CA .

17. May 26, 2014

### smakhtar

Hey, I finally figured it out. Thanks for the help. I did BC x AC and I got it right, which is the same as |BC||AC|sin (theta) right?

18. May 26, 2014

### Nathanael

Yes that is the same, but the correct answer is half of that.

Also you made a mistake right here:
2-3 = -1, not 1 (could've been a typo, but still it could've affected your answer)

19. May 26, 2014

### SammyS

Staff Emeritus
That's the area of the parallelogram.

Divide by 2 to get the area of the triangle.

20. May 26, 2014

I did.