11.3-bo'lim : Nuqta mahsuloti
10. Berilgan \(\vec u = 7\vec i - \vec j + \vec k\) va \(\vec w = - 2\vec i + 5\vec j - 6\vec k\) \({{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u\) ni hisoblashadi.
Yechim
Bu erda biz qilishimiz kerak bo'lgan yagona narsa eslatmalardan formuladan foydalanishdir. Buning uchun quyidagi miqdorlar kerak bo'ladi.
\[\vec u\centerdot \vec w = - 25\hspace{0.25in}\hspace{0.25in}{\left\| {\vec w} \right\|^2} = 65\]Keyin proyeksiya,
\[{{\mathop{\rm proj}\nolimits} _{\,\vec w}}\,\vec u = \frac{{ - 25}}{{65}}\left( { - 2\vec i + 5\vec j - 6\vec k} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{10}}{{13}}\vec i - \frac{{25}}{{13}}\vec j + \frac{{30}}{{13}}\vec k}}\]