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Question
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (−6, 3), B (−2, −5)
Find \[\left| \vec{A} B \right|\] in each case.
Solution
Given: \[A\left( - 6, 3 \right)\] and \[B \left( - 2, - 5 \right)\]
Then, the position vector \[\vec{AB}\] is given by \[\vec{AB} =\] Position vector of B - Position vector of A
\[= \left( - 2 \hat{i} - 5 \hat{j} \right) - \left( - 6 \hat{i} + 3 \hat{j} \right)\]
\[ = - 2 \hat{i} - 5 \hat{j} + 6 \hat{i} - 3 \hat{j} \]
\[ = 4 \hat{i} - 8 \hat{j}\]
So,
\[\left| \vec{AB} \right| = \sqrt{4^2 + \left( - 8 \right)^2} = \sqrt{16 + 64} = \sqrt{80} = 4\sqrt{5}\]
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