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Answer on Question #54705, Math Calculus
Draw diagrams to show two vectors a and b, and the vectors a + b and a − b.
When is the magnitude of a + b less than that of a − b?
When is the magnitude of a + b equal to that of a − b?
When is |a + b| = |a| + |b|?
Solution
Fig.1 a  b
Fig.2 a  b
See Fig.1:
a b 
a  b  2 a  b cos  
2
a  b  2a  b 
2
2
2
(1)
where  is the angle between vectors a and  b .
See Fig.2:
a b 
 
2
a  b  2a  b 
2
a  b  2 a  b cos    
2
2
(2)
a  b  a  b  a  b  a  b  a  b  2 a  b cos    a  b  2 a  b cos     
2
2
2
2
2
2
2 a  b cos    2 a  b cos      cos    cos      cos     cos    2 cos    0 
cos    0     0,  / 2 
The magnitude of a  b less than that of a  b if    0,  / 2  .
a  b  a  b  a  b  a  b  a  b  2 a  b cos    a  b  2 a  b cos     
2
2
2
2
2
2
2 a  b cos    2 a  b cos      cos    cos      cos     cos    2 cos    0 
cos    0     / 2
The magnitude of a  b equal to that of a  b if    / 2
2

a b  a  b

2
 a  b  2 a  b cos    a  b  2 a  b  2 a  b cos   
2
2
2
2
 2 a  b  cos    1    0
a  b  a  b if vectors a and b are parallel.
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