Answer to Question #141897 in Analytic Geometry for Vasantha

Question

Find the semi vertical angle of a cone with axis x/1=y/2=z/3 and x-axis is the generator?

Answer 1

$\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\text{ direction vector }\vec{a}(1,2,3)$

$\text{axis }X\text{ direction vector }\vec{i}(1,0,0)$

$\text{The desired angle is the angle between straight lines or their direction vectors}$

$\cos{\varphi}=\frac{\vec{a}*\vec{i}}{\vert{\vec{a}}\vert*\vert{\vec{i}}\vert}=\frac{1}{1*\sqrt{1+4+9}}$

$\cos{\varphi}=\frac{1}{\sqrt{13}};{\varphi}\approx74\degree$

Answer:$cos{\varphi}=\frac{1}{\sqrt{13}};{\varphi}\approx74\degree$