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Show that two non-zero vectors \( \vec{A}=A_{1} \hat{i}+A_{2} \hat{j}+A_{3} \hat{k} \) and \( \vec{B}=B_{1} \hat{i}+B_{2} \hat{j}+B_{3} \hat{k} \) are parallel if and only if \( \frac{A_{1}}{B_{1}}=\frac{A_{2}}{B_{2}}=\frac{A_{3}}{B_{3}} \) |
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Answer» Vector \(\vec A\) and \(\vec B\) are parallel if and only if \(\vec A\times\vec B=0\) \(\Leftrightarrow\)\(\begin{vmatrix}\hat i&\hat j&\hat k\\A_1&A_2&A_3\\B_1&B_2&B_3\end{vmatrix}=0\) \(\Leftrightarrow\) \(\hat i\) (A2B3 - A3B2) + \(\hat j\) (B1A3 - B3A1) + \(\hat k\) (A1B2 - B1A2) = 0 = 0\(\hat i\) + 0\(\hat j\) + 0\(\hat k\) \(\Leftrightarrow\) A2B3 - A3B2 = 0, B1A3 - B3A1 = 0 and A1B2 - B1A2 = 0 \(\Leftrightarrow\) \(\frac{A_2}{B_2}=\frac{A_3}{B_3},\frac{A_1}{B_1}=\frac{A_3}{B_3}\) and \(\frac{A_1}{B_1}=\frac{A_2}{B_2}\) \(\Leftrightarrow\) \(\frac{A_1}{B_1}=\frac{A_2}{B_2}=\frac{A_3}{B_3}\) Hence Proved. |
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