Calculating the Angle Between Two Vectors

In summary, to calculate the angle between the vectors A and B, we first find the dot product A*B by multiplying the corresponding components and adding them together. Then, we find the magnitude of A and B by taking the square root of the sum of the squares of their components. Dividing the dot product by the product of the magnitudes gives us the cosine of the angle, which we can then find by taking the inverse cosine. In this case, the angle between A and B is approximately 78.46 degrees.
  • #1
needhelp83
199
0
Calculate the angle between the vectors:
A = 6.8i + 4.5j + 6.2k
B = 8.2i + 2.3j – 7.0k


A*B= AB cos
A*B=AxBx + AyBy + AzBz
A*B= (6.8)(8.2) + (4.5)(2.3) + (6.2)(-7.0)=22.71
22.71 cos = 87.48 degrees

Correct?
 
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  • #2
needhelp83 said:
Calculate the angle between the vectors:
A = 6.8i + 4.5j + 6.2k
B = 8.2i + 2.3j – 7.0k


A*B= AB cos
A*B=AxBx + AyBy + AzBz
A*B= (6.8)(8.2) + (4.5)(2.3) + (6.2)(-7.0)=22.71
22.71 cos = 87.48 degrees

Correct?

What exactly did you write here? A.B=|A||B|cos(A,B) implies cos(A,B) = (A.B) / (|A||B|).
 
  • #3
radou said:
What exactly did you write here? A.B=|A||B|cos(A,B) implies cos(A,B) = (A.B) / (|A||B|).


I am not too sure what you mean by the A.B and |A||B|
 
  • #4
A.B means A "dot" B
This is a dot product and conceptually means to multiply two vectors only in the same direction as each other (for example, if you push on a cart downward, and it goes forward because of the angle of your force, the dot product would be the magnitude (that's what the lines mean) of the force in the direction of the distance (that's why Cos@) times the magnitude of the distance) or you could do magnitude of distance in direction of force times magnitude of force.
 
  • #5
needhelp83 said:
Calculate the angle between the vectors:
A = 6.8i + 4.5j + 6.2k
B = 8.2i + 2.3j – 7.0k


A*B= AB cosθ
A*B=AxBx + AyBy + AzBz
A*B= (6.8)(8.2) + (4.5)(2.3) + (6.2)(-7.0)=22.71
22.71 cos = 87.48 degrees

Correct?
The 22.71 part is good, but where did you calculate the AB in AB cosθ?
 
  • #6
needhelp83 said:
I am not too sure what you mean by the A.B and |A||B|

By |A||B| I meant what you denoted as AB - the product of the absolute values of the vectors.
 
  • #7
A=6.8+4.5+6.2=17.5
B=8.2+2.3-7.0=3.5

AB=61.25

22.71/61.25=cos theta=68.28 degrees

I think it should be 78.40 degrees because I found this dot product calculator, but I am not sure how they go to this answer
 
  • #8
needhelp83 said:
A=6.8+4.5+6.2=17.5
B=8.2+2.3-7.0=3.5
The components of a vector are perpendicular--you can't just add them to find the magnitude!

How you find the magnitude of a vector from its components?
 
  • #9
needhelp83 said:
A=6.8+4.5+6.2=17.5
B=8.2+2.3-7.0=3.5

AB=61.25

22.71/61.25=cos theta=68.28 degrees

I think it should be 78.40 degrees because I found this dot product calculator, but I am not sure how they go to this answer
That is not correct for AB
A = sqrt(A*A)
B = sqrt(B*B)
where I have used your symbol * for the dot product.

What you did for A*B was perfect. Do it again for A*A and B*B To find A and B and their product.
 
  • #10
A=sqrt(AxAx + AyAy + AzAz)=sqrt((6.8)(6.8) + (4.5)(4.5) + (6.2)(6.2))=10.24

B=sqrt(BxBx + ByBy + BzBz)=sqrt((8.2)(8.2) + (2.3)(2.3) + (-7.0)(-7.0))=11.02

AB=(10.24)(11.02)=112.85

22.71/112.85=0.20 cos theta= 78.46 degrees

Think I finally got it. Correct?
 
  • #11
needhelp83 said:
A=sqrt(AxAx + AyAy + AzAz)=sqrt((6.8)(6.8) + (4.5)(4.5) + (6.2)(6.2))=10.24

B=sqrt(BxBx + ByBy + BzBz)=sqrt((8.2)(8.2) + (2.3)(2.3) + (-7.0)(-7.0))=11.02

AB=(10.24)(11.02)=112.85

22.71/112.85=0.20 cos theta= 78.46 degrees

Think I finally got it. Correct?

Looks good. :smile:
 

FAQ: Calculating the Angle Between Two Vectors

What is the definition of "angle between vectors"?

The angle between vectors is the angle formed between two vectors in a vector space. It is the measure of the difference in direction of the two vectors.

How is the angle between vectors calculated?

The angle between vectors can be calculated using the dot product formula: θ = cos⁻¹(a · b / |a||b|), where a and b are the two vectors and |a| and |b| are their magnitudes.

What is the range of values for the angle between vectors?

The angle between vectors can range from 0° (when the two vectors are parallel) to 180° (when the two vectors are antiparallel). It can also be negative if the vectors are pointing in opposite directions.

What is the significance of the angle between vectors?

The angle between vectors can provide information on the relationship between the two vectors. If the angle is close to 0°, it means the vectors are similar or pointing in the same direction. If the angle is close to 180°, it means the vectors are opposite or pointing in opposite directions. It can also be used to determine the orthogonality of the vectors (if the angle is 90°).

How is the angle between vectors used in real-world applications?

The concept of angle between vectors is used in various fields such as physics, engineering, and computer graphics. It is used to calculate forces and velocities in physics, determine the orientation of objects in engineering, and to create realistic animations in computer graphics.

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