Question

Given a cube $ABCD{A}_1{B}_1{C}_1{D}_1$ with lower base $ABCD$, 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 centres of the faces $ABCD$ and $A_1{B}_1{C}_1{D}_1$ respectively. $O$ is a point on the line $M{M}_1$ such that $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=\overrightarrow{O{M}_1},\overrightarrow{OM}=\lambda \overrightarrow{O{M}_1}$ if $\lambda$ is equal to

• A. 1/16
• B. 1/8
• C. 1/4
• D. 1/2

Answer 1

Answer: (C)

$\overrightarrow{O{M}_1}=\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}$
$=\overrightarrow{OM}+\overrightarrow{MA}+\overrightarrow{OM}+\overrightarrow{MB}+\overrightarrow{OM}+\overrightarrow{MC}+\overrightarrow{OM}+\overrightarrow{MD}$
$=4\overrightarrow{OM}$
$\overrightarrow{OM}=(1/4){\overrightarrow{OM}}_1$
$\lambda =\frac{1}{4}$