question 1
The nonzero vector U and V are orthogonal if and only if U x V = 0 .

question 2
U x V = 0 if and only if U and V are orthogonal.

Can you explain to me why question 1 is false ?
I know question 2 is true because it is the definition of orthogonal.

Thank you very much.

2. Originally Posted by Jenny20
question 1
The nonzero vector U and V are orthogonal if and only if U x V = 0 .

question 2
U x V = 0 if and only if U and V are orthogonal.

Can you explain to me why question 1 is false ?
I know question 2 is true because it is the definition of orthogonal.

Thank you very much.
False, False, False.
Two vectors are othrogonal if and only if their dot product is zero.

3. Hello, Jenny!

1) The nonzero vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ are orthogonal if and only if $\displaystyle \vec{u} \times \vec{v}\:=\:0$

The answer is false . . . Right!

Two vectors are orthogonal if their dot product is zero: .$\displaystyle \vec{u}\cdot\vec{v}\,=\,0$

2) $\displaystyle \vec{u} \times \vec{v}\:=\:0$ if and only if $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ are orthogonal.
.
Strange . . . This is identical to question 1.

$\displaystyle \vec{u} \times \vec{v}\,=\,0 \;\;\longleftrightarrow\;\;\vec{u} \parallel \vec{v}$

The cross product is zero if and only if the vectors are parallel.

4. Question 2 is not identical to Question 1. The assumption that the vectors are nonzero has been dropped.

It's still false though for essentially the same reasons that Question 1 is false.

5. $\displaystyle \begin{array}{l} I: < 1,0,0 > \quad \& \quad J: < 0,1,0 > \\ I \cdot J = 0\quad \& \quad I \times J = < 0,0,1 > \\ \end{array}$
The above shows that #1 is false.

$\displaystyle \begin{array}{l} A: < 2,0,0 > \quad \& \quad B: < 1,0,0 > \\ A \times B = < 0,0,0 > \quad \& \quad A \cdot B \not= 0 \\ \end{array}.$
The above shows that #2 is false.

6. I see.