Why does the vector | A.B | = | a |. | B | (and if and only if a is parallel to B, the equal sign holds)

Why does the vector | A.B | = | a |. | B | (and if and only if a is parallel to B, the equal sign holds)

a. B = ||||||||||||||||||||||||||||||||||||

It is proved that for any vector a, B has | a | - | B | ≤| A-B | ≤| a | + | B |

It's the trilateral relationship of a triangle

a. If B is a vector, then (a × b) × a · B = (a × b) & # 178; is it true? Proof
It's point by point

(a × b) is a vector, and (a × b) × A is a vector
(a × b) × a · B can only be [(a × b) × a] · B
[（a×b）×a]·b=(（a×b）,a,b)=(a,b,（a×b）)=（a×b）·（a×b）=（a×b）².

For any vector a, B proves that |||||- |||||≤||||||||||||||||||||≤||||||||!

Proof: 1, because, for any vector a, B, for any vector a, B, B has: a + B 124; 124; 124| 124124; 124; 124124124124\124\124\\\\124\\\\| (B-A-A + the proof of | A-B | a - | B | A-B | is proved by (1)