# vectors: are these valid?

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• Sep 24th 2007, 11:52 AM
shilz222
vectors: are these valid?
If $\displaystyle \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$ does it follow that $\displaystyle \vec{b} = \vec{c}$?

I said that $\displaystyle \vec{a} \cdot (\vec{b}- \vec{c}) = 0$ which means that they are perpendicular which implies that $\displaystyle \vec{b} \neq \vec{c}$.

If $\displaystyle \vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ does it follow that $\displaystyle \vec{b} = \vec{c}$?

I said that $\displaystyle \vec{a} \times (\vec{b} - \vec{c}) = 0$ which means that they are parallel, and so $\displaystyle \vec{b} \neq \vec{c}$.
• Sep 24th 2007, 12:11 PM
Jhevon
Quote:

Originally Posted by shilz222
If $\displaystyle \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$ does it follow that $\displaystyle \vec{b} = \vec{c}$?

I said that $\displaystyle \vec{a} \cdot (\vec{b}- \vec{c}) = 0$ which means that they are perpendicular which implies that $\displaystyle \vec{b} \neq \vec{c}$.

actually, this means the vector $\displaystyle \vec {a}$ is perpendicular to the vector $\displaystyle \vec{b} - \vec {c}$. it does not say anything about the vectors $\displaystyle \vec{b}$ and $\displaystyle \vec{c}$ or how they relate to each other.

Quote:

If $\displaystyle \vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ does it follow that $\displaystyle \vec{b} = \vec{c}$?

I said that $\displaystyle \vec{a} \times (\vec{b} - \vec{c}) = 0$ which means that they are parallel, and so $\displaystyle \vec{b} \neq \vec{c}$.
a similar observation can be made here
• Sep 24th 2007, 12:22 PM
Plato
Because both are false, find counter examples, such as:
$\displaystyle A = \left\langle {1,1,1} \right\rangle ,\;B = \left\langle {1,0, - 1} \right\rangle ,\;C = \left\langle { - 1,1,0} \right\rangle ,\;\& \;A \cdot B = A \cdot C$.