Understanding Vector Dot Product: Solving for a, b, and c in a Right Triangle

[RadiationX]

i have three vectors: a=4,b=3,c=5 that form a right triangle.
vector a is in the positive x direction, vector b is in the positive y direction starting at the tip of a. vector c is the hypotenuse of the triangle with tip at the origin. (see attaced picture .doc file)


the questions are: what is a dot b, a dot c, and b dot c.


i have the solutions in my manual but i don't understand them.


the manual says that from the figure it is clear that a + b + c = 0, where a is perpindicular to b:


a dot b = 0, since the angle between them is 90 degrees:


a dot c = a dot (-a-b)=-|a|^2=-16

and similarly b dot c = -9

i have no idea whay this is true. any help would be appreciated especially a general explanition of what the dot product is

 

[dextercioby]

Is that a joke...?You posted only 2...

Daniel.

 

[cronxeh]

A dot product between vector a and vector b is this:

a dot b = |a|*|b|*cos(theta)
a dot b = (AxBx)i + (AyBy)j + (AzBz)k

Those are two definitions and they are equal. In your case, you are given the length of vectors (a=4,b=3,c=5). This is the called the magnitude of a vector. the |a| = 4, |b| = 3, |c| = 5. How would you find the angle theta between the two vectors?

There are two laws you can use. Law of sines and law of cosines. Or a pythogoras' theorem if the vectors form a right angle. In your case the triangle is right, because 4^2 + 3^2 = 5^2. So you can use a good old SOH CAH TOA rule (Sin = Opposite/Hypothenus, Cos = Adjacent/Hypothenus, Tan = Opposite/Adjecent).

Try to visualize the triangle first. Obviously c is a hypothenus with length 5.

This is given: |a| = 4, |b| = 3, |c| = 5
And you want to find:
1] a dot b = |a|*|b|*cos(theta)= 4*3*cos(90) = 0
2] a dot c = |a|*|c|*cos(theta) = 4*5*4/5 = 16
3] b dot c = |b|*|c|*cos(theta) = 3*5*3/5 = 9

 

[RadiationX]

ok i made my mistake with soh cah toa , but the last two answers are -16 and -9. why the negative sign? thx

 

[Doc Al]

why the negative sign?

Because the angle between the vectors ($\theta$) is 90 < $\theta$ < 180 degrees, a region in which $cos \theta$ is negative.

 

[RadiationX]

dot product

Because the angle between the vectors ($\theta$) is 90 < $\theta$ < 180 degrees, a region in which $cos \theta$ is negative.

i'm not saying that your're wrong but how do you know that cos is in the second quad? from the picture this is not obvious.

 

[Doc Al]

To find the angle between two vectors, redraw them so that their tails start at the same point. Direction matters!

 

[RadiationX]

that's it! thank you Doc AI