Find vectors making a certain angle with given vectors

In summary, the homework statement asks for all unit vectors in R4 making an angle of pi/3 with the three vectors A=(1,1,-1,-1) B=(1,-1,1,-1) and C=(1,-1,-1,1). Using V=(w,x,y,z) as the vector we are trying to find, I solved the above equation for all three vectors A,B and C obtaining simultaneous equations to solve, however I obtained x=y=z and w2-wx-x2=0. I am not sure how to solve this to find all unit vectors. Also is there a method of doing this that I am unaware of?How did you get w2-
  • #1
LASmith
21
0

Homework Statement



Find all unit vectors in R4 making an angle of [itex]\pi/3[/itex] with the three vectors
A=(1,1,-1,-1)
B=(1,-1,1,-1)
C=(1,-1,-1,1)

Homework Equations


u.v=|u||v|cos[itex]\Theta[/itex]

The Attempt at a Solution


using V=(w,x,y,z) as the vector we are trying to find, I solved the above equation for all three vectors A,B and C obtaining simultaneous equations to solve, however I obtained
x=y=z and w2-wx-x2=0
I am not sure how to solve this to find all unit vectors.
Also is there a method of doing this that I am unaware of?
 
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  • #2
How did you get w2-wx-x2=0? Did you use the fact that V is a unit vector?
 
  • #3
vela said:
How did you get w2-wx-x2=0? Did you use the fact that V is a unit vector?

I haven't yet considered the unit vector part, I was just going to divide the answer by the modulus.

I obtained this results as |u|cos[itex]\Theta[/itex] is 2*1/2=1 for all the vectors A,B&C therefore right hand side of the equation for all the vectors is |v|, so we get v.u=|v| and I took

|v| = [itex]\sqrt{}w2+x2+y2+z2[/itex]

I put C.u=B.u to obtain w-x-y+z=w-x+y-z which gave me that y=z

Then A.u=B.u to obtain w+x-y-z=w-x+y-z which gave me x=y

Then A.u=C.u to give w+x-y-z=w-x-y+z which gives x=z

so if you take x, y & z to all equal x (for example)

and use any of the equations we get w-x=[itex]\sqrt{}w2+3x2[/itex]
from this I got the w2-wx-x2=0
 
  • #4
LASmith said:
and use any of the equations we get [itex]w-x = \sqrt{w^2+3x^2}[/itex].
from this I got the w2-wx-x2=0.
Recheck your algebra. The second equation doesn't follow from the first.

Also, if you use the fact that |V|=1, the first equation will give you w=1+x.
 
Last edited:
  • #5
vela said:
Recheck your algebra. The second equation doesn't follow from the first.

Also, if you use the fact that |V|=1, the first equation will give you w=1+x.


Okay, sorry I got the algebra wrong, I do not obtain w2-wx-x2=0 instead, I get w=-x. Which would be a solution, however, then fact vela pointed out w=1+x contradicts this, so what value am I supposed to use?
 
  • #6
How'd you get w=-x? I got wx+x2=0. Try again!
 
  • #7
vela said:
How'd you get w=-x? I got wx+x2=0. Try again!

Yes, I obtained this too, the you get x(w+x)=0
Therefore you get either x=0 or w+x=0 which leads to my answer w=-x
 
  • #8
Heh. I didn't recognize the root. But there's no problem. You have two equations and two unknowns:
\begin{align*}
w+x & = 0 \\
w-x & = 1
\end{align*}
You can solve that.

Also, don't just ignore the x=0 solution as well.
 
  • #9
vela said:
w+x & = 0 \\
w-x & = 1
\end{align*}
You can solve that.

Also, don't just ignore the x=0 solution as well.

therefore we get w=1/2 therefore x=-1/2 y=-1/2 and z=-1/2

Also using x=0 we get w=0, y=0 & z=0
so when it asks for all the unit vectors there must only be one, as the second one has a modulus of zero. So are they no more unit vectors apart from
(0.5,-0.5,-0.5,-0.5)?
 
  • #10
You didn't handle the x=0 case correctly. In particular, it doesn't follow that w=0 from x=0.
 
  • #11
vela said:
In particular, it doesn't follow that w=0 from x=0.

Yes, I see it now, if x=0, then the |v|=1 still so we can use the equation w-x=1.
This implies that w=1, and as x=y=z=0
We get the second unit vector (1,0,0,0)

So there are two unit vectors for this answer. Thank you for all your help :)
 

1. How do I find the angle between two given vectors?

To find the angle between two given vectors, first calculate the dot product of the two vectors. Then, divide the dot product by the product of the magnitudes of the two vectors. Finally, take the inverse cosine of this value to find the angle in radians.

2. Can I find a vector that makes a specific angle with two given vectors?

Yes, you can find a vector that makes a specific angle with two given vectors by using the dot product formula and some simple trigonometry. First, calculate the dot product of the two given vectors. Then, use the dot product, the angle you want the new vector to make with one of the given vectors, and the magnitude of the given vectors to find the magnitude of the new vector. Finally, use the magnitude and angle to calculate the components of the new vector.

3. Is it possible to find multiple vectors that make a certain angle with given vectors?

Yes, it is possible to find multiple vectors that make a certain angle with given vectors. This can be achieved by using the dot product formula and solving for the magnitude and components of the new vector for different angles.

4. How can I use vectors to determine the orientation of an object?

Vectors can be used to determine the orientation of an object by finding the angle between the object's orientation vector and a reference vector. This can be done by using the dot product formula and taking the inverse cosine of the resulting value.

5. Can I use vectors to determine the direction of motion in a system?

Yes, vectors can be used to determine the direction of motion in a system by calculating the velocity and acceleration vectors of each object in the system. The direction of motion can be determined by the angle between these vectors.

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