Answer to Question #217144 in Analytic Geometry for joe

Question

Find the value(s) of x such that the angle between the vectors (0, 1, −1) and (−1, x, 0) is 2π/3 . Show all your calculations

Answer 1

Solution

We know ,if x and y are two vectors ,the angle $\theta$ between them is given by;

Cos $\theta$ =$\frac{(\vec{x}.\vec{y})}{\Vert x\Vert\vert y\Vert}$

Cos($\frac{2π}{3}$)=-$\frac12$

$\vec{x}.\vec{y}$ =0(-1)+x (1)+0(-1)=x

$\Vert \vec{x}\Vert$ =$\sqrt{0^2+1^2+(-1)^2}$ =$\sqrt{2}$

$\Vert\vec{y}\Vert$ =$\sqrt{(-1)^2+(x)^2+0^2}$ =$\sqrt{1+x^2}$

We now equate;

$-\frac 12$ =$\frac{x}{\sqrt{2(1+x^2)}}$

2(1+x²)=4x²

x²=1

x=$\sqrt{1}$ =1 or -1

If x =1

Cos $\theta$ =$\frac 12$ $\neq$ Cos$(\frac{2π}{3})$

Chose X=-1 that gives the cos$\theta$ as $-\frac12$

Answer

x=-1