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Addition of Vectors

Given a cube ...

Question

Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with lower base $A B C D$ , upper base $A_{1} B_{1} C_{1} D_{1}$ and the lateral edges $A A_{1}, B B_{1}, C C_{1}$ and $D D_{1}$ ; $M$ and $M_{1}$ are the centers of the faces $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ respectively. $O$ is apoint on line $M M_{1}$ , such that
$O A + O B + O C + O D = O M_{1}$ , then $O M = λ O M_{1}$ is $λ =$

\frac{1}{4}

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\frac{1}{2}

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\frac{1}{6}

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\frac{1}{8}

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Solution

The correct option is A $\frac{1}{4}$
$O M_{1} = O A + O B + O C + O D$ (given)
$= O M + M A + O M + M B + O M + M C + O M + M D$
$= 4 O M + (M A + M C) + (M B + M D) = 4 O M$ $(∵ M A = - M C, M B = - M D)$
$∴ O M = \frac{1}{4} O M_{1},$
$∴ λ = \frac{1}{4}$

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