Question

Calculate the scalar product of the following vectors.
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 lateral edges $A{A}_1,B{B}_1,C{C}_1,andD{D}_1.$ Prove that on the straight line $\left(M{M}_1\right),$ where M and $M_1$ are the centres of the faces ABCD and $A_1{B}_1{C}_1{D}_1,$ respectively, there is a point O such that $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}=O{M}_1.$ show that $\overrightarrow{MO}=\lambda O{M}_{1}andfind\lambda .$

Answer 1

Answer: $\lambda =\frac{1}{4}$