Calculating the Angle Between Two Vectors

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Homework Help Overview

The discussion revolves around calculating the angle between two vectors using the dot product and magnitudes of the vectors. The vectors in question are A = 6.8i + 4.5j + 6.2k and B = 8.2i + 2.3j – 7.0k.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants explore the calculation of the dot product and its relation to the angle between the vectors. There are attempts to clarify the notation used for the dot product and the magnitudes of the vectors. Questions arise about the correct interpretation of the calculations and the steps involved in finding the angle.

Discussion Status

The discussion includes various calculations and interpretations of the dot product and magnitudes. Some participants provide corrections and guidance on how to properly calculate the magnitudes of the vectors, while others express uncertainty about their calculations. There is no explicit consensus on the final angle, but there are productive exchanges regarding the methodology.

Contextual Notes

Participants question the assumptions made in the calculations, particularly regarding the addition of vector components to find magnitudes. There is also confusion about the notation and the steps required to derive the angle from the dot product.

needhelp83
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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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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|).
 
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|
 
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.
 
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?
 
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.
 
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
 
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?
 
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:
 

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