Answer to Question #200635 in Analytic Geometry for Vina

Question

Assume that U is a plane.

Find out whether or not the following vectors lie in U:

(10.1) ~u =< 3.8, 1 >, ~v =< −4, 1, 1 > and w~ = −~v

(10.2) ~u =< 3.8, 1 >, ~v =< −4, 1, 1 > and w~ = ~u − ~

Answer 1

Vectors lie in U if

$u\cdot(v\times w)=0$

1.

$v\times w=\begin{vmatrix} i & j&k \\ -4 & 1&1\\ 4&-1&-1 \end{vmatrix}=0$

All vectors are in the plane U.

2.

$v\times w=\begin{vmatrix} i & j&k \\ -4 & 1&1\\ -3&-8&-1 \end{vmatrix}=7i-7j+35k$

$u\cdot(v\times w)=(3,8,1)\cdot(7,-7,35)=21-56+35=0$

All vectors are in the plane U.