Advertisements (Quảng cáo)
Bài 2. Cho tam giác \(ABC\) có hai điểm \(M,N\) sao cho
\(\left\{ \matrix{
\overrightarrow {AM} = \alpha \overrightarrow {AB} \hfill \cr
\overrightarrow {AN} = \beta \overrightarrow {AC} \hfill \cr} \right.\)
a) Hãy vẽ \(M, N\) khi \(\alpha = {2 \over 3};\beta = – {2 \over 3}\)
b) Hãy tìm mối liên hệ giữa \(α, β\) để \(MN//BC\)
a) Ta có:
\(\eqalign{
& \overrightarrow {AM} = {2 \over 3}\overrightarrow {AB} \Leftrightarrow \left\{ \matrix{
\overrightarrow {AM} \uparrow \uparrow \overrightarrow {AB} \hfill \cr
AM = {2 \over 3}AB \hfill \cr} \right. \cr
& \overrightarrow {AN} = – {2 \over 3}\overrightarrow {AC} \Leftrightarrow \left\{ \matrix{
\overrightarrow {AN} \uparrow \downarrow \overrightarrow {AC} \hfill \cr
AN = {2 \over 3}AC \hfill \cr} \right. \cr} \)
b) Ta có:
\(\eqalign{
& \overrightarrow {AM} = \alpha \overrightarrow {AB} \cr
& \overrightarrow {AN} = \beta \overrightarrow {AC} \cr
& \Rightarrow \overrightarrow {AM} – \overrightarrow {AN} = \alpha \overrightarrow {AB} – \beta \overrightarrow {AC} \cr
& \Rightarrow \overrightarrow {MN} = \alpha \overrightarrow {AB} – \beta \overrightarrow {AC} \cr
& \Rightarrow \overrightarrow {MN} = \alpha (\overrightarrow {AB} – {\beta \over \alpha }\overrightarrow {AC} ),\alpha \ne 0 \cr} \)
Ta cũng có: \(\overrightarrow {BC} = – (\overrightarrow {AB} – \overrightarrow {AC} )\)
Do đó, để \(MN // BC\) thì
\(\left\{ \matrix{
\overrightarrow {AB} – \overrightarrow {AC} \hfill \cr
\overrightarrow {AB} – {\beta \over \alpha }\overrightarrow {AC} \hfill \cr} \right.\)
phải cùng phương, cho ta \({\beta \over \alpha } = 1 \Rightarrow \alpha = \beta \)