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NOTE: I know the solution, but I challenge it. Specifically, I feel that the generally accepted solution to this problem
is not intuitive. I have shown the accepted solution, and below that I have shown what I feel seems more
intuitive (from a geometric standpoint)
1. The problem statement
find the three angles of the triangle with vertices (2 1 1) (1 3 5) (3 4 4).
Dot Product: A⋅B = A B cosθ
Analysis:
define the points:
ptA = (2 1 1)
ptB = (1 3 5)
ptC = (3 4 4)
create (position) vectors wrt origin (0,0,0):
A = <20,10,10> = <2, 1, 1>
B = <1, 3, 5>
C = <3, 4, 4>
define the angles:
α (between vectors A&B) (alpha)
β (between vectors B&C) (beta)
γ (between vectors C&A) (gamma)
apply the dot product of each (position) vector wrt to one another:
A*B = norm(A)*norm(B)*cos(α) ∴ α = 90.00°
B*C = norm(B)*norm(C)*cos(β) ∴ β = 22.49°
C*A = norm(A)*norm(B)*cos(γ) ∴ γ = 67.50°
Please view: https://www.physicsforums.com/forums/calculusandbeyondhomework.156/attachments/acceptedsolutionpleasecheckanglesjpg.77050/?temp_hash=21a6e771909902657d7fa4fa3b0d6ae7 [Broken]
1. The problem statement
AS ABOVE SO BELOW
Dot Product: A⋅B = A B cosθ
Vector Addition: B = A + (B  A)
Flowchart: define the points:
create (position) vectors wrt origin (0,0,0):
find displacement vectors:
start point is point A, followed by B, C, and back to A.
then employ vector addition (final minus initial)
then employ vector addition of the entire "displacementvectored" triangle
define the angles & Please view second image:
apply the dot product of each (displacement) vector wrt to one another:
Analysis:
*skipped to step 3 of Flowchart*
find displacement vectors:
"path": Start at ptA, the ptB, ptC and then return to ptA
displacement vectors: B = A + (B  A)
=> dBA = B  A
=> dCB = C  B
=> dAC = A  C
triangle described by vector sum: dBA + dCB = dAC
define the angles then solve: (Please see image: https://www.physicsforums.com/forums/calculusandbeyondhomework.156/attachments/myunderstandingofthevectorspng.77059/?temp_hash=21a6e771909902657d7fa4fa3b0d6ae7 [Broken])
α = acosd(sum(dAC.*dCB)*inv(norm(dAC)*norm(dCB))) ∴ α = 90.00°
β = acosd(sum(dBA.*dAC)*inv(norm(dBA)*norm(dAC))) ∴ β = 157.5085°
γ = acosd(sum(dCB.*dBA)*inv(norm(dCB)*norm(dBA))) ∴ γ = 112.4915°
Adjustments: (I am off by 180°)
adjusted_alpha = 180°  abs(alpha)
=> adjusted_alpha = 90.00°
=> adjusted_beta = 22.49°
=> adjusted_gamma = 67.50°
END
Intuitive Explanation anyone?
How is my geometry wrong? What intuitive (geometric) reasons justify subtracting off 180°. Also, I am ignoring the sign of the angle, hence the abs operator above..I don't feel this is an effective solution. Can you please help me?
HAPPY NEW YEAR!
is not intuitive. I have shown the accepted solution, and below that I have shown what I feel seems more
intuitive (from a geometric standpoint)
1. The problem statement
find the three angles of the triangle with vertices (2 1 1) (1 3 5) (3 4 4).
Homework Equations
Dot Product: A⋅B = A B cosθ
The Attempt at a Solution
Analysis:
define the points:
ptA = (2 1 1)
ptB = (1 3 5)
ptC = (3 4 4)
create (position) vectors wrt origin (0,0,0):
A = <20,10,10> = <2, 1, 1>
B = <1, 3, 5>
C = <3, 4, 4>
define the angles:
α (between vectors A&B) (alpha)
β (between vectors B&C) (beta)
γ (between vectors C&A) (gamma)
apply the dot product of each (position) vector wrt to one another:
A*B = norm(A)*norm(B)*cos(α) ∴ α = 90.00°
B*C = norm(B)*norm(C)*cos(β) ∴ β = 22.49°
C*A = norm(A)*norm(B)*cos(γ) ∴ γ = 67.50°
Please view: https://www.physicsforums.com/forums/calculusandbeyondhomework.156/attachments/acceptedsolutionpleasecheckanglesjpg.77050/?temp_hash=21a6e771909902657d7fa4fa3b0d6ae7 [Broken]
1. The problem statement
AS ABOVE SO BELOW
Homework Equations
Dot Product: A⋅B = A B cosθ
Vector Addition: B = A + (B  A)
The Attempt at a Solution
Flowchart: define the points:
create (position) vectors wrt origin (0,0,0):
find displacement vectors:
start point is point A, followed by B, C, and back to A.
then employ vector addition (final minus initial)
then employ vector addition of the entire "displacementvectored" triangle
define the angles & Please view second image:
apply the dot product of each (displacement) vector wrt to one another:
Analysis:
*skipped to step 3 of Flowchart*
find displacement vectors:
"path": Start at ptA, the ptB, ptC and then return to ptA
displacement vectors: B = A + (B  A)
=> dBA = B  A
=> dCB = C  B
=> dAC = A  C
triangle described by vector sum: dBA + dCB = dAC
define the angles then solve: (Please see image: https://www.physicsforums.com/forums/calculusandbeyondhomework.156/attachments/myunderstandingofthevectorspng.77059/?temp_hash=21a6e771909902657d7fa4fa3b0d6ae7 [Broken])
α = acosd(sum(dAC.*dCB)*inv(norm(dAC)*norm(dCB))) ∴ α = 90.00°
β = acosd(sum(dBA.*dAC)*inv(norm(dBA)*norm(dAC))) ∴ β = 157.5085°
γ = acosd(sum(dCB.*dBA)*inv(norm(dCB)*norm(dBA))) ∴ γ = 112.4915°
Adjustments: (I am off by 180°)
adjusted_alpha = 180°  abs(alpha)
=> adjusted_alpha = 90.00°
=> adjusted_beta = 22.49°
=> adjusted_gamma = 67.50°
END
Intuitive Explanation anyone?
How is my geometry wrong? What intuitive (geometric) reasons justify subtracting off 180°. Also, I am ignoring the sign of the angle, hence the abs operator above..I don't feel this is an effective solution. Can you please help me?
HAPPY NEW YEAR!
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