ABCD is a parallelogram.

Question

$ABCD$ is a parallelogram. The position vectors of $A$ and $C$ are respectively $3\hat i+3\hat j+5\hat k$ and $\hat i-5\hat j-5\hat k$. If $M$ is the midpoint of the diagonal $DB$, then the magnitude of the projection of $\overrightarrow{OM}$ on $\overrightarrow{OC}$, where $O$ is the origin, is:

Accepted Answer

In a parallelogram, diagonals bisect each other, so the midpoint of $DB$ equals the midpoint of $AC$:

$$M = \frac{(3+1)\hat i+(3-5)\hat j+(5-5)\hat k}2 = 2\hat i-\hat j$$

$\overrightarrow{OC}=\hat i-5\hat j-5\hat k$, $|\overrightarrow{OC}|=\sqrt{51}$; $\overrightarrow{OM}=2\hat i-\hat j$, $|\overrightarrow{OM}|=\sqrt5$.

$$\text{Projection} = \frac{\overrightarrow{OM}\cdot\overrightarrow{OC}}{|\overrightarrow{OC}|} = \frac{2+5+0}{\sqrt{51}}=\frac7{\sqrt{51}}$$

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