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Question
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (4, −1), B (1, 3)
Find \[\left| \vec{A} B \right|\] in each case.
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Solution
Given: \[A (4, - 1)\] and \[B\left( 1, 3 \right)\]
Then the position vector \[\vec{AB}\] is given by \[\vec{AB}\] = Position vector of B - Position vector of A \[= \left( \hat{i} + 3 \hat{j} \right) - \left( 4 \hat{i} - \hat{j} \right)\]
\[ =\hat{i} + 3 \hat{j} - 4 \hat{i} + \hat{j} \]
\[ = - 3 \hat{i} + 4 \hat{j} \]
So,
\[\left| \vec{AB} \right| = \sqrt{\left( - 3 \right)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
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