Cho bốn điểm bất kì M, N, P, Q. Chứng minh các đẳng thức sau:
\(\begin{array}{l}
a)\overrightarrow {PQ} + \overrightarrow {NP} + \overrightarrow {MN} = \overrightarrow {MQ} \\
b)\overrightarrow {NP} + \overrightarrow {MN} = \overrightarrow {QP} + \overrightarrow {MQ} \\
c)\overrightarrow {MN} + \overrightarrow {PQ} = \overrightarrow {MQ} + \overrightarrow {PN}
\end{array}\)
a) Ta có
\(\begin{array}{l}
\overrightarrow {PQ} + \overrightarrow {NP} + \overrightarrow {MN} \\
= \left( {\overrightarrow {MN} + \overrightarrow {NP} } \right) + \overrightarrow {PQ} \\
= \overrightarrow {MP} + \overrightarrow {PQ} = \overrightarrow {MQ}
\end{array}\)
(đpcm)
b)
\(\begin{array}{*{20}{l}}
\begin{array}{l}
\overrightarrow {NP} + \overrightarrow {MN} \\
= \left( {\overrightarrow {NQ} + \overrightarrow {QP} } \right) + \left( {\overrightarrow {MQ} + \overrightarrow {QN} } \right)
\end{array}\\
\begin{array}{l}
= \overrightarrow {QP} + \overrightarrow {MQ} + \overrightarrow {NQ} + \overrightarrow {QN} \\
= \overrightarrow {QP} + \overrightarrow {MQ}
\end{array}
\end{array}\)
c)
\(\begin{array}{*{20}{l}}
\begin{array}{l}
\overrightarrow {MN} + \overrightarrow {PQ} \\
= \left( {\overrightarrow {MQ} + \overrightarrow {QN} } \right) + \left( {\overrightarrow {PN} + \overrightarrow {NQ} } \right)
\end{array}\\
\begin{array}{l}
= \overrightarrow {MQ} + \overrightarrow {PN} + \overrightarrow {QN} + \overrightarrow {NQ} \\
= \overrightarrow {MQ} + \overrightarrow {PN}
\end{array}
\end{array}\)
-- Mod Toán 10
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