$A B C D$ is 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 $D B$, then the magnitude of the projection of $\overrightarrow{O M}$ on $\overrightarrow{O C}$, where $O$ is…

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$A B C D$ is 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 $D B$, then the magnitude of the projection of $\overrightarrow{O M}$ on $\overrightarrow{O C}$, where $O$ is the origin, is

Aicharya · Accepted Answer

In a parallelogram, diagonals bisect each other. So, mid point of $D B$ is also the midpoint of $A C$. Mid-point of $M=2 \hat{i}-\hat{j}$ Direction ratio of $O C=(1,-5,-5)$ Direction ratio of $O M=(2,-1,0)$ Angle $\theta$ between $O M$ and $O C$ is given by $ \left.\cos \theta=\frac{(1 \times 2+)-5(-1)(+-5) 0(}{\sqrt{2^2+(-1)^2} \sqrt{()^2+(-5)^2+(-5)^2}}\right)() $ $ =\frac{2+5}{\sqrt{5} \sqrt{51}}=\frac{7}{\sqrt{5} \sqrt{51}} $ Projection of $\overrightarrow{O M}$ on $\overrightarrow{O C}$ is given by $ |O M| \cdot \cos \theta=\sqrt{5} \times \frac{7}{\sqrt{5} \times \sqrt{51}}=\frac{7}{\sqrt{51}} $

$A B C D$ is parallelogram. The position vectors of $A$ and $C$ are respectively, $3 \hat{i}+3 \hat{j}+5…

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