Vectors a, b, c make 60^(circ) angles with each other. |a|=4, |b|=2, |c|=6. Find the length of p=a+b+c.

Hayley Bernard

Hayley Bernard

Answered question

2022-07-17

Vectors a, b, c make 60 angles with each other. | a | = 4, | b | = 2 , | c | = 6. Find the length of p=a+b+c.
The only way I can think of a, b and c having 60 angles with each other is that they form a vertex of a tetrahedron. Then, I can find |a+b| or |b+c| or |a+c| using the law of cosines. But then I can't find |p|, because I don't know the angle between the vector I have found and the remaining one.
I would like to get some hints or clues how to solve this, thanks in advance.

Answer & Explanation

eishale2n

eishale2n

Beginner2022-07-18Added 15 answers

I would square the given sum:
p 2 = ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 a b + 2 b c + 2 c a
This is equal
| a | 2 + | b | 2 + | c | 2 + 2 | a | | b | cos ( π / 3 ) + 2 | b | | c | cos ( π / 3 ) + 2 | a | | c | cos ( π / 3 )

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