ข้อสอบ A-Level คณิตประยุกต์ 1 ปี 2559 (วิชาสามัญเก่า)

$$\begin{gathered} \mathrm{จากโจทย์} (\mathrm{ก}.) \left(\overline{\mathrm{u}}\times \overline{\mathrm{v}}\right)·\overline{\mathrm{w}}=\overline{\mathrm{u}}·\left(\overline{\mathrm{v}}\times \overline{\mathrm{w}}\right) \\ -\mathrm{พิจารณาฝั่งขวา} \\ \overline{\mathrm{u}}·\left(\overline{\mathrm{v}}\times \overline{\mathrm{w}}\right)=\overline{\mathrm{v}}·\left(\overline{\mathrm{w}}\times \overline{\mathrm{u}}\right)=\overline{\mathrm{w}}·\left(\overline{\mathrm{u}}\times \overline{\mathrm{v}}\right)=\left(\overline{\mathrm{u}}\times \overline{\mathrm{v}}\right)·\overline{\mathrm{w}}\hspace{0.17em} \\ \mathrm{ดังนั้น}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{ข้อ} \mathrm{ก}. \mathrm{ถูก} \end{gathered}$$

$$\begin{gathered} \mathrm{จากโจทย์} (\mathrm{ข}.) \left(\overline{\mathrm{u}}\times \overline{\mathrm{v}}\right)\times \overline{\mathrm{w}}=\overline{\mathrm{u}}\times \left(\overline{\mathrm{v}}\times \overline{\mathrm{w}}\right) \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \\ -\mathrm{กำหนดให้} \overline{\mathrm{v}}=\overline{\mathrm{i}}, \overline{\mathrm{v}}=\overline{\mathrm{i}},\overline{ \mathrm{w}}=\overline{\mathrm{j}} \\ \mathrm{จะได้} \left(\overline{\mathrm{i}}\times \overline{\mathrm{i}}\right)\times \overline{\mathrm{j}}=\overline{\mathrm{i}}\times \left(\overline{\mathrm{i}}\times \overline{\mathrm{j}}\right) \\ \overline{0}\times \overline{\mathrm{j}}=\overline{\mathrm{i}}\times \overline{\mathrm{k}} \\ \overline{0}=\overline{\mathrm{i}}\times \overline{\mathrm{k}} \\ \hspace{0.17em} \\ \mathrm{ดังนั้น}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{ข้อ} \mathrm{ข}. \mathrm{ผิด} \end{gathered}$$

$$\begin{gathered} \mathrm{จากโจทย์} (\mathrm{ค}.) \left(\overline{\mathrm{u}}-\overline{\mathrm{v}}\right)·\left(\overline{\mathrm{u}}+\overline{\mathrm{v}}\right)={\left|\overline{\mathrm{u}}\right|}^2-{\left|\overline{\mathrm{v}}\right|}^2 \\ -\mathrm{พิจารณาฝั่งซ้าย} \\ \left(\overline{\mathrm{u}}-\overline{\mathrm{v}}\right)·\left(\overline{\mathrm{u}}+\overline{\mathrm{v}}\right)=\overline{\mathrm{u}}·\overline{\mathrm{u}}+\cancel{\overline{\mathrm{u}}·\overline{\mathrm{v}}}-\cancel{\overline{\mathrm{v}}·\overline{\mathrm{u}}}-\overline{\mathrm{v}}·\overline{\mathrm{v}} \\ ={\left|\overline{\mathrm{u}}\right|}^2-{\left|\overline{\mathrm{v}}\right|}^2 \hspace{0.17em}\hspace{0.17em} \\ \mathrm{ดังนั้น}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{ข้อ} \mathrm{ค}. \mathrm{ถูก} \end{gathered}$$

$$\begin{gathered} \mathrm{จากโจทย์} (\mathrm{ง}.) \left(\overline{\mathrm{u}}-\overline{\mathrm{v}}\right)\times \left(\overline{\mathrm{u}}+\overline{\mathrm{v}}\right)=2\left(\overline{\mathrm{u}}\times \overline{\mathrm{v}}\right) \\ -\mathrm{พิจารณาฝั่งซ้าย} \\ \left(\overline{\mathrm{u}}-\overline{\mathrm{v}}\right)\times \left(\overline{\mathrm{u}}+\overline{\mathrm{v}}\right)=\overline{\mathrm{u}}\times \overline{\mathrm{u}}+\overline{\mathrm{u}}\times \overline{\mathrm{v}}-\overline{\mathrm{v}}\times \overline{\mathrm{u}}-\overline{\mathrm{v}}\times \overline{\mathrm{v}} \hspace{0.17em}\hspace{0.17em};\overline{\mathrm{u}}\times \overline{\mathrm{v}}=-\overline{\mathrm{v}}\times \overline{\mathrm{u}} \\ =\overline{0}+\overline{\mathrm{u}}\times \overline{\mathrm{v}}+\overline{\mathrm{u}}\times \overline{\mathrm{v}} -\overline{0} \\ =2\left(\overline{\mathrm{u}}\times \overline{\mathrm{v}}\right) \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \\ \mathrm{ดังนั้น}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{ข้อ} \mathrm{ง}. \mathrm{ถูก} \end{gathered}$$