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Question
Find the position vector of the mid-point of the vector joining the points P (2, 3, 4) and Q(4, 1, −2).
Solution
Let \[\vec{p} , \vec{q}\] be the position vectors of the points \[P\left( 2, 3, 4 \right), Q\left( 4, 1, - 2 \right)\]
Then,
\[\vec{p} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}\] and \[\vec{q} = 4 \hat{i} + \hat{j} - 2 \hat{k}\] Therefore, the position vector of the midpoint of the given points is \[\frac{\vec{p} + \vec{q}}{2}\]
∴\[\frac{\vec{p} + \vec{q}}{2} = \frac{(2 \hat{i} + 3 \hat{j} + 4 \hat{k} ) + (4 \hat{i} + \hat{j} - 2 \hat{k} )}{2} = 3 \hat{i} + 2 \hat{j} + \hat{k}\]
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