Coplanarity

Geometry Level 2

Are points $A(1,2,4)$ , $B(8,16,32)$ , $C(64,128,256)$ and $D(512,1024,2048)$ coplanar ?

Yes Cannot be determined No

2 solutions

Tom Engelsman
May 8, 2021

Let $O(0,0,0)$ be the origin in $\mathbb{R^{3}}$ . If we have the vector $OA = \hat{i} + 2\hat{j} + 4\hat{k}$ , then:

$OB = 8 \cdot OA$ ,

$OC = 64 \cdot OA = 8 \cdot OB$ ,

$OD = 512 \cdot OA = 64 \cdot OB = 8 \cdot OC.$

Each vector $OB, OC, OD$ is scalar multiple of $OA \Rightarrow$ all four points are collinear $\Rightarrow$ all four points are coplanar.

Steven Chase
Dec 9, 2017

They are colinear, and colinearity implies coplanarity.

This is true. It also makes solving the problem very easy.

Marta Reece - 3 years, 3 months ago