Question about determining the angles of triangle given two vectors

[RoboNerd]

<<Mentor note: Missing template due to originally being posted elsewhere>>

Hello everyone.

I have the following problem:

Determine the angles of a triangle where two sides of a triangle are formed by the vectors

A = 3i -4j -k and B=4i -j + 3k

I thought that I would find the third side being represented by vector R which would be equal to A-B, that is
R = -i -3j -4k.

I would then take the cross products of each combination of the two vectors and find the angle between them and these would be the angles of the triangle.

Dot producting A and B vectors, I get the angle between them to be 60 degrees.

Dot producting the A and R vectors, I also get 60 degrees.

Thus with the final angle = 180 - 60 - 60, the last angle should be 60 degrees also.

This is blatantly wrong as my solutions tell me that the answer is:
arcos 7 /sqrt(75), arcos sqrt(26)/sqrt(75), 90 degrees, or 36degrees4', 53degrees56', 90 degrees

Could anyone please direct me as to what I did wrong and what mistakes need to be fixed? Thanks in advance.

 

[Orodruin]

Your method and results are fine.

 

[Ray Vickson]

<<Mentor note: Missing template due to originally being posted elsewhere>>

Hello everyone.

I have the following problem:

Determine the angles of a triangle where two sides of a triangle are formed by the vectors

A = 3i -4j -k and B=4i -j + 3k

I thought that I would find the third side being represented by vector R which would be equal to A-B, that is
R = -i -3j -4k.

I would then take the cross products of each combination of the two vectors and find the angle between them and these would be the angles of the triangle.

Dot producting A and B vectors, I get the angle between them to be 60 degrees.

Dot producting the A and R vectors, I also get 60 degrees.

Thus with the final angle = 180 - 60 - 60, the last angle should be 60 degrees also.

This is blatantly wrong as my solutions tell me that the answer is:
arcos 7 /sqrt(75), arcos sqrt(26)/sqrt(75), 90 degrees, or 36degrees4', 53degrees56', 90 degrees

Could anyone please direct me as to what I did wrong and what mistakes need to be fixed? Thanks in advance.

Your solution is correct.

However, a slightly easier way might have been to compute the lengths |A|, |B| and |R| = |A-B|; you would find these to all be equal, so your triangle is equilateral.

 

[RoboNerd]

Thank you everyone for your inputs. It is a relief to know that I did not mess something up.