向量题：设|a|=3|b|=4|c|=5,且满足a+b+c=0,则|a叉乘b+b叉乘c+c叉乘a|是多少?

Question

向量题：设|a|=3 |b|=4 |c|=5,且满足a+b+c=0 ,则 |a叉乘b+b叉乘c+c叉乘a| 是多少?

Answer 1

|a|=3,|b|=4,|c|=5,a+b+c=0
a²+b²=c² 故向量 a、b、c 构成一个直角三角形,斜边为 c,直角边为 a 和 b
| a x b + b x c + c x a |
=| a x b + b x (-a-b)+ (-a-b) x a |
=| a x b - b x a - b x b - a x a - b x a |
=| a x b + a x b + a x b |
=3 | a x b |
=3 |a| |b| sinC
=3 x 3 x 4 x sin90°
=36