Matematické Fórum

stuart clark · 02. 05. 2015 08:36

In a parallelogram $\bf{ABCD}\;,$ If $\vec{AB} = a,\vec{AD}=b$ and $\vec{AC} = c.$ Then $\vec{DB}\cdot \vec{AB}$ has a value

$\bf{Options::}$

$(1)\; \frac{1}{2}\left(a^2+b^2+c^2\right)$

$(2)\; \frac{1}{2}\left(b^2+c^2-a^2\right)$

$(3)\; \frac{1}{2}\left(c^2+a^2-b^2\right)$

$(4)\; \frac{1}{2}\left(a^2+b^2-c^2\right)$

Brano · 12. 05. 2015 23:11 — Editoval Brano (12. 05. 2015 23:12)

$\vec{c}=\vec{a}+\vec{b}$ and $\vec{DB}=\vec{a}-\vec{b}$
thus $c^2=a^2+b^2+2\vec{a}\cdot\vec{b}$ and
$\vec{DB}\cdot \vec{AB}=\vec{a}(\vec{a}-\vec{b})=\frac{1}{2}(3a^2+b^2-c^2)$
so
(1) holds iff $a^2=c^2$
(2) holds iff $2a^2=c^2$
(3) holds iff $\vec{a}\cdot\vec{b}=0$
(4) holds iff $\vec{a}=0$

all of them are possible, but none is general.

stuart clark · 11. 07. 2015 12:51

Thanks ↑ Brano:

Matematické Fórum

#1 02. 05. 2015 08:36

Vectors

#2 12. 05. 2015 23:11 — Editoval Brano (12. 05. 2015 23:12)

Re: Vectors

#3 11. 07. 2015 12:51

Re: Vectors

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