Determine Right Angle of ΔABC with Vector Methods

[masterofthewave124]

Q: Given the points A(1,6,-2), B(2,5,3) and C(5,3,2). Use two different vector methods to determine whether ΔABC is a right angle triangle.

hmmm vector methods...i'm not quite sure what to do but i have some ideas. for a first method, could i take the magnitude of each side and see if the trig calculations hold valid (pretty weak method considering there's many assumptions here). I'm thinkign a better method would be dot and cross product. i.e. for dot product, see if any two sides produces a product of zero?

i would just someone to guide me the right way so no time is wasted guessing around.

thanks in advance!

 

[arunbg]

Hint: Use dot product for 1 method and cross product for the second, longer, method . Start by forming vectors AB, BC and CA .

 

[HallsofIvy]

Q: Given the points A(1,6,-2), B(2,5,3) and C(5,3,2). Use two different vector methods to determine whether ΔABC is a right angle triangle.

hmmm vector methods...i'm not quite sure what to do but i have some ideas. for a first method, could i take the magnitude of each side and see if the trig calculations hold valid (pretty weak method considering there's many assumptions here).

Trig calculations? Do you mean sine and cosine? Since you don't know any angles I don't see how you could do that. Of course the Pythagorean theorem might help.

I'm thinkign a better method would be dot and cross product. i.e. for dot product, see if any two sides produces a product of zero?

Yes, that would work as a second method.

i would just someone to guide me the right way so no time is wasted guessing around.

thanks in advance!

Time is never wasted while "guessing around". The more you do it the more you learn.

 

[masterofthewave124]

for dot product: how do i know which two vectors could potentially have an angle of 90 between them? would i have to try all 3 possibilities?

for cross product: what am i doing here?

HallsofIvy: what i meant for trig calculations was in fact pythag. to see if any two sides produced a third side that was the sum of the squares of the other two (i.e. a^2 + b^2 = c^2)

you're also right about guessing around, that's how we learn i guess.

 

[Hootenanny]

You have basically been given three vector equations in the question;

$$\vec{OA} = 1i + 6j -2k$$

$$\vec{OB} = 2i + 5j -3k$$

$$\vec{OC} = 1i + 6j -2k$$

You now need to obtain three more vector equations; $\vec{AB}$, $\vec{BC}$, $\vec{CA}$. Can you do that?

For the dot product basically, your just going to have to try all three combinations (there are only three).

More information on the cross product can be found at
http://en.wikipedia.org/wiki/Cross_product"
http://mathworld.wolfram.com/CrossProduct.html"

 

[arunbg]

You can also try Pythagoras theorem using the magnitudes of the vectors as Hallsofivy explained .

 

[HallsofIvy]

You are given A(1,6,-2), B(2,5,3) and C(5,3,2). So $\vec{AB}= (2-1)\vec{i}+ (5-6)\vec{j}+ (-2-3)\vec{k}$, $\vec{AC}= (1-5)\vec{i}+ (6-3)\vec{j}+ (-2-2)\vec{k}$, and $\vec{BC}= (5-2)\vec{i}+ (3-5)\vec{j}+ (2-3)\vec{i}$. Is the dot product of any two of those 0? The cross product isn't applicable here.

 

[arunbg]

The cross product isn't applicable here.

Why not ? If triangle ABC is rt angled at B , then ,
$$\vert(\vec{AB}\times\vec{BC})\vert = \vert\vec{AB}\vert \vert\vec{BC}\vert$$.
This is surely a necessary and sufficient condition although a bit long .

 

[Hootenanny]

Why not ? If triangle ABC is rt angled at B , then ,
$$\vert(\vec{AB}\times\vec{BC})\vert = \vert\vec{AB}\vert \vert\vec{BC}\vert$$.
This is surely a necessary and sufficient condition although a bit long .

I believe the cross product gives the vector which is perpendicular to both A and B.

 

[arunbg]

I was only taking the magnitude (modulus). Also sin(B) = 1 .

 

[masterofthewave124]

yeah the cross product would work here...none of dot products produced a value of 0 so i guess its not right-angled. didn't make an arithmetic error did i?

edit:

wait if i used the cross product here, then a x b = |a||b| for any vector combination as arunbg stated. but then the end result would be (x,y,z) = a number; how would i rationalize the equality there?

 

[arunbg]

yeah the cross product would work here...none of dot products produced a value of 0 so i guess its not right-angled. didn't make an arithmetic error did i?
Perhaps you did make an arithmetic mistake. Also note that in Hallsofivy's last post , vector AB is given wrongly . It should have been
$\vec{AB}= (2-1)\vec{i}+ (5-6)\vec{j}+ (2+3)\vec{k}$
Note the change in the last term. Now can you do it ?

but then the end result would be (x,y,z) = a number; how would i rationalize the equality there?

Just take the modulus(magnitude) of (x,y,z) as I have shown in my earlier post .

Arun

 

[masterofthewave124]

yeah i noticed HallsOfIvy's mistake when reading it, i figured it was a typo. but even with the corrected values, i can't get it to work.

these are the values I am getting:
AB • BC
= 10

AB • CA
= -27

BC • CA
= -14

i think it has something to do with the vectors I am forming; i formed AB, BC and CA but HallsofIvy formed AB, BC and AC. I am getting confused because dot product is taken from tip to tip anyways.

 

[Hootenanny]

Are you sure you are calculating the dot product correctly? Here's my working for the first one;

$$\vec{AB}= (2-1)\vec{i}+ (5-6)\vec{j}+ (2+3)\vec{k}$$

$$\vec{BC}= (5-2)\vec{i}+ (3-5)\vec{j}+ (2-3)\vec{k}$$

$$\vec{AB}{\mathbf\centerdot}\vec{BC} = 1\times 3 + -1 \times -2 + 5\times -1$$

$$\boxed{\vec{AB}{\mathbf\centerdot}\vec{BC} = 0}$$

Now what does this say about the vectors $\vec{AB}$ and $\vec{BC}$? If you can't see it straight away, note that the cosine of the angle between them is given by;

$$\cos\theta = \frac{\vec{AB}{\mathbf\centerdot}\vec{BC}}{|\vec{AB}||\vec{BC}|}$$

 

[masterofthewave124]

sorry I'm so careless (did 3+2+5 instead of 3+2-5). thanks hootenay (note that you have a mistake in your dot product, the first value should be 1 x 3 but i know it was a typo since your answer is still 0).

 

[Hootenanny]

sorry I'm so careless (did 3+2+5 instead of 3+2-5). thanks hootenay (note that you have a mistake in your dot product, the first value should be 1 x 3 but i know it was a typo since your answer is still 0).

Ahh yes, careless error, I seem to be prone to them recently. Thank you for the correction. Now, have you thought anymore about my question;

Now what does this say about the vectors $\vec{AB}$ and $\vec{BC}$?

 

[masterofthewave124]

it says that the angle between the vectors AB and BC is 90 and hence a right-angle. is this what you were implying?

 

[Hootenanny]

it says that the angle between the vectors AB and BC is 90 and hence a right-angle. is this what you were implying?

That's it. So now you've proved it with one vector method...