Angle Between Two Vectors: Solving for θ

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Homework Help Overview

The problem involves finding the angle between two vectors given their scalar and vector products. The scalar product is -7.00, and the magnitude of the vector product is 3.00. The context is rooted in vector mathematics, specifically dealing with trigonometric relationships in the context of angles between vectors.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the calculation of the angle using the tangent function derived from the sine and cosine relationships. Questions arise regarding the expected units for the angle (degrees or radians) and the range of the arctangent function. There is exploration of the implications of the negative cosine value on the quadrant in which the angle lies.

Discussion Status

The discussion is active, with participants providing insights into the properties of the arctangent function and its range. There is an ongoing examination of how to correctly interpret the angle based on the quadrant and the values derived from the tangent function. Guidance has been offered regarding the need to consider the correct quadrant for the angle based on the sign of the cosine.

Contextual Notes

Participants note the importance of understanding the range of the arctangent function and its implications for determining the correct angle. The original poster expresses confusion about the results obtained from their calculations, indicating a need for clarification on the angle's quadrant and the correct interpretation of the trigonometric functions involved.

nabelekt
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Homework Statement



Vectors and have scalar product -7.00 and their vector product has magnitude 3.00.
What is the angle between these two vectors?

Homework Equations



|A| |B| cosθ = -7
|A| |B| sinθ = 3

The Attempt at a Solution



tanθ = (-3/7)
tan-1(-3/7)=θ

When I enter tan-1(-3/7) into my calculator, I get -23.199, but when I enter 23.199 into MasteringPhysics, it tells me that I am incorrect. What am I doing wrong?

Thanks!
 
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Is the angle expected in degrees or radians?
 


The answer is required to be in degrees.
 


What is the range of arctan? From the given values, can you determine whether the angle is really in that range?
 


I believe that the range of arctan is 270°<θ<90°. Correct? 23.199° seems to be in that range…

Thanks for the help. It's been a while since I've done this stuff.
 


The range given does not make sense. The range should be (-π/2,π/2).
 


I guess it would actually lie in quadrant II whereas the range of arctan is quadrants I and IV. So what should I do?
 


You should express you angle via some auxiliary angle that is in the range of arctan, obtain the value of the auxiliary angle, and then get your angle.
 
  • #10


nabelekt said:

Homework Statement



Vectors and have scalar product -7.00 and their vector product has magnitude 3.00.
What is the angle between these two vectors?

Homework Equations



|A| |B| cosθ = -7
|A| |B| sinθ = 3

The Attempt at a Solution



tanθ = (-3/7)
tan-1(-3/7)=θ

When I enter tan-1(-3/7) into my calculator, I get -23.199, but when I enter 23.199 into MasteringPhysics, it tells me that I am incorrect. What am I doing wrong?

Thanks!

You should take care to notice two things here:

(1) If |A| |B| cosθ = -7, then cosθ must be negative. In which quadrants is cosθ negative?

(2) tanθ has a period of ∏ radians, so tanθ = tan(θ+n∏) for any integer n, and tan-1(tanθ) = θ +n∏. Can you find a value of n that gives you an angle in the correct quadrant for θ +n∏=tan-1(-3/7)?
 

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