Finding the angle of unit vectors

In summary, to find the angle between \vec{OA} and \vec{OB} given that they are unit vectors, we can use the law of cosines. By rearranging the equation in terms of cos(A), we can solve for the angle without using the scalar product. The law of cosines states that a^2 = b^2 + c^2 - 2bccosA, and since the length is 1, b=c. With this information, we can solve for cos(A) and then find the angle using inverse cosine.
  • #1
rbnphlp
54
0
If [itex]\vec{OA}[/itex] is the unit vector [itex]l_1i+m_1j+n_1k[/itex] and [itex]\vec{OB}[/itex] is the unit vector [itex]l_2i+m_2j+n_2k[/itex], by using the cosine formula in triangle OAB find the angle between [itex]\vec{OA}[/itex] &[itex]\vec{OB}[/itex]..

I have tried expressing them as direction cosines , but none of that is working..can anyone point me in the right direction..
Also I have not been introduced to scalar product ..is there an easy way of go about doing this without using the scalar product
Thanks
 
Last edited:
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  • #2
Hint: Use the law of cosines.
 
  • #3
D H said:
Hint: Use the law of cosines.

the question does state to use that ..a further hint maybe:(
 
  • #4
What is the law of cosines?
 
  • #5
D H said:
What is the law of cosines?

a^2=b^2+c^2-2bccosA ..

and since the length is 1 b=c

but what do I do with a unit vector?
 
  • #6
rbnphlp said:
a^2=b^2+c^2-2bccosA ..
You are trying to solve for cos(A), not a^2. You already know (or can know) a^2. Why don't you re-arrange the above in terms of solving for cos(A)?
 
  • #7
D H said:
You are trying to solve for cos(A), not a^2. You already know (or can know) a^2. Why don't you re-arrange the above in terms of solving for cos(A)?

thanks ...I have solved it now
 

Related to Finding the angle of unit vectors

1. What is the definition of a unit vector?

A unit vector is a vector with a magnitude of 1 and is often used to indicate direction. It is represented by a lowercase letter with a hat ( ̂) on top, such as ̂a.

2. How do you find the angle between two unit vectors?

To find the angle between two unit vectors, you can use the dot product formula: θ = cos-1 (a · b), where a and b are the two unit vectors. This will give you the angle in radians.

3. What is the relationship between unit vectors and the Cartesian coordinate system?

Unit vectors are the basis of the Cartesian coordinate system. The unit vectors i, j, and k represent the x, y, and z directions respectively. Any vector in the Cartesian coordinate system can be expressed as a linear combination of these unit vectors.

4. How do you convert a vector to a unit vector?

To convert a vector to a unit vector, you can divide the vector by its magnitude. This will result in a vector with the same direction, but a magnitude of 1.

5. Can unit vectors be negative?

Yes, unit vectors can be negative. The negative sign indicates the direction of the vector, while the magnitude remains 1. For example, - ̂a represents a unit vector in the opposite direction of ̂a.

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