First slide

Algebra of vectors

Question

If $\vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0, where \vec{a}, \vec{b} and \vec{c}$ are non-coplanar, then

Moderate

A

$\vec{r} ⊥ (\vec{c} \times \vec{a})$
B

$\vec{r} ⊥ (\vec{a} \times \vec{b})$
C

$\vec{r} ⊥ (\vec{b} \times \vec{c})$
D

$\vec{r} = \vec{0}$

Solution

Let $\vec{r} \neq \vec{0} . Then \vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = 0$
hence $\vec{a}, \vec{b} and \vec{c}$ are coplanar, which is a contradiction.
Therefore, $\vec{r} = \vec{0}$

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