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Question
Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the through the points (0, 3, 2) and (3, 5, 6).
Solution
Suppose vector
\[\overrightarrow{a}\] is passing through the points (1, -1 , 2) and (3, 4 , -2 ) and
\[\overrightarrow{a}\] is passing through the points (0, 3, 2) and (3, 5, 6).
Then,
\[\overrightarrow{a} = 2 \hat{i} + 5 \hat{j} - 4\stackrel\frown{k} \]
\[ \overrightarrow{b} = 3 \hat{i} + 2 \hat{j} + 4 \stackrel\frown{k} \]
Now,
\[\overrightarrow{a} . \overrightarrow{b} = \left( 2 \hat{i} + 5 \hat{j} - 4 \stackrel\frown{k} \right) . \left( 3 \hat{i} + 2 \hat{j} + 4 \stackrel\frown{k} \right) = 0\]
Hence, the given lines are perpendicular to each other.
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