Show that the scalar product of two vectors obeys the distributive law.
(N/A) According to the figure,let $\overrightarrow{OP} = \vec{A}$,$\overrightarrow{OQ} = \vec{B}$,and $\overrightarrow{QR} = \vec{C}$.
Then,$\overrightarrow{OR} = \overrightarrow{OQ} + \overrightarrow{QR} = \vec{B} + \vec{C}$.
The scalar product $\vec{A} \cdot (\vec{B} + \vec{C})$ is defined as the product of the magnitude of $\vec{A}$ and the projection of $(\vec{B} + \vec{C})$ onto the direction of $\vec{A}$.
From the figure,the projection of $\vec{B}$ on $\vec{A}$ is $OM$,and the projection of $\vec{C}$ on $\vec{A}$ is $MN$.
Therefore,the projection of $(\vec{B} + \vec{C})$ on $\vec{A}$ is $ON = OM + MN$.
Now,$\vec{A} \cdot (\vec{B} + \vec{C}) = |\vec{A}| (ON) = |\vec{A}| (OM + MN)$.
$= |\vec{A}| (OM) + |\vec{A}| (MN)$.
Since $|\vec{A}| (OM) = \vec{A} \cdot \vec{B}$ and $|\vec{A}| (MN) = \vec{A} \cdot \vec{C}$,we have:
$\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$.
This proves that the scalar product obeys the distributive law.
Then,$\overrightarrow{OR} = \overrightarrow{OQ} + \overrightarrow{QR} = \vec{B} + \vec{C}$.
The scalar product $\vec{A} \cdot (\vec{B} + \vec{C})$ is defined as the product of the magnitude of $\vec{A}$ and the projection of $(\vec{B} + \vec{C})$ onto the direction of $\vec{A}$.
From the figure,the projection of $\vec{B}$ on $\vec{A}$ is $OM$,and the projection of $\vec{C}$ on $\vec{A}$ is $MN$.
Therefore,the projection of $(\vec{B} + \vec{C})$ on $\vec{A}$ is $ON = OM + MN$.
Now,$\vec{A} \cdot (\vec{B} + \vec{C}) = |\vec{A}| (ON) = |\vec{A}| (OM + MN)$.
$= |\vec{A}| (OM) + |\vec{A}| (MN)$.
Since $|\vec{A}| (OM) = \vec{A} \cdot \vec{B}$ and $|\vec{A}| (MN) = \vec{A} \cdot \vec{C}$,we have:
$\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$.
This proves that the scalar product obeys the distributive law.
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