Vector true/False

[KLS2111]

Hello,
I am trying to figure out if for any two vectors u and v, if u.(u x v)=0 always? I can't seem to find a counter example, but that doesn't mean that one doesn't exist. If anyone can let me know if I am correct in my thinking that this statement is true and/or why I would appreciate it more than my method of guess and check. Thank you!

 

[Deleted member 4993]

Hello,
I am trying to figure out if for any two vectors u and v, if u.(u x v)=0 always? I can't seem to find a counter example, but that doesn't mean that one doesn't exist. If anyone can let me know if I am correct in my thinking that this statement is true and/or why I would appreciate it more than my method of guess and check. Thank you!

yes - you can prove it directly by assigning

u = ai + bj + ck

v = xi + yj + zk

Then u.(u x v) would be a scalar whose magnitude is determinant of:

|a   b   c|
|a   b   c|
|x   y   z|

The determinant of matrix above is 0 - because two rows are dependant.

 

[soroban]

Hello, KLS2111!

\(\displaystyle \text{Prove that: }\:\vec u\cdot\left(\vec u \times \vec v\right)\:=\:0\;\text{ for any two vectors }\vec u\text{ and }\vec v.\)

A sketch might help . . .

                |
                | u x v
                |
            *---|---------------*
           /    |   /          /
          /     |  /          /
         /      | /v         /
        /       |/   u      /
       /        *-------   /
      *-------------------*

\(\displaystyle \vec u \times \vec v\text{ is a vector }perpendicular\text{ to the plane of }\vec u\text{ and }\vec v.\)

. . \(\displaystyle \text{Hence, }\vec u\text{ and }(\vec u \times \vec v)\text{ are perpendicular.}\)


\(\displaystyle \text{And the dot product of two perpendicular vectors is zero: }\;\vec u \cdot (\vec u \times \vec v) \;=\;0\)

 

[KLS2111]

Thanks! The diagram makes so much sense to me!