Answer:
The angle between [tex](2,2,4)[/tex] and [tex](2,-1,1)[/tex] is 60º.
Step-by-step explanation:
From linear algebra, we can determine the angle between both vectors by definition of dot point:
[tex]\cos \theta = \frac{\vec u\bullet \vec v}{\|\vec u\|\cdot \| \vec v\|}[/tex] (1)
Where:
[tex]\vec u[/tex], [tex]\vec v[/tex] - Vectors.
[tex]\|\vec u\|[/tex], [tex]\|\vec v\|[/tex] - Norms of vectors.
[tex]\theta[/tex] - Angle between vectors, measured in sexagesimal degrees.
If we know that [tex]\vec u = (2,2,4)[/tex] and [tex]\vec v = (2,-1,1)[/tex], then angle between vectors is:
[tex]\|\vec u\| = \sqrt{\vec u\bullet \vec u}[/tex] (2)
[tex]\|\vec u\| = \sqrt{2^{2}+2^{2}+4^{2}}[/tex]
[tex]\|\vec u\| \approx 4.899[/tex]
[tex]\|\vec v\| = \sqrt{\vec v\bullet \vec v}[/tex] (3)
[tex]\|\vec v\| = \sqrt{2^{2}+(-1)^{2}+1^{2}}[/tex]
[tex]\|\vec v\| \approx 2.450[/tex]
[tex]\vec u \bullet \vec v = (2,2,4)\bullet (2,-1,1)[/tex]
[tex]\vec u \bullet \vec v = 4-2+4[/tex]
[tex]\vec u \bullet \vec v = 6[/tex]
[tex]\cos \theta = \frac{6}{(4.899)\cdot (2.450)}[/tex]
[tex]\cos \theta = 0.5[/tex]
[tex]\theta = 60^{\circ}[/tex]
The angle between [tex](2,2,4)[/tex] and [tex](2,-1,1)[/tex] is 60º.
