Respuesta :
Answer:
The Euclidean distance between these two vectors is [tex]E = \sqrt{219}[/tex].
[tex]\cos{\alpha} = -\frac{4}{10\sqrt{215}}[/tex]
The angle is obtuse.
Step-by-step explanation:
Suppose we have two vectors, u and v, in which:
[tex]u = (x_{0}, y_{0}, z_{0}, w_{0})[/tex]
[tex]v = (x_{1}, y_{1}, z_{1}, w_{1})[/tex]
The Euclidean distance between these two vectors can be given by the following formula:
[tex]E = \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2} + (z_{1} - z_{0})^{2} + (w_{1} - w_{0})^{2}}[/tex]
To find the cosine of the angle [tex]\alpha[/tex] between two vectors u and v, we use the following formula:
[tex]\cos{\alpha} = \frac{u.v}{||u||||v||}[/tex]
In which [tex]u.v[/tex] is the dot product between vectors u and v, [tex]||u||[/tex] is the norm of vector u and [tex]||v||[/tex] is the norm of vector v.
If [tex]\cos{\alpha}[/tex] is negative, the angle is obtuse. If it is 0, the angle is 90º. And if it is positive, the angle is acute.
In this problem, we have that:
[tex]u = (-8,-4,6,3)[/tex]
[tex]||u|| = \sqrt{(-8)^{2} + (-4)^{2} + (6)^{2} + (3)^{2}} = \sqrt{125} = 5\sqrt{5}[/tex]
[tex]v = (5,3,6,4)[/tex]
[tex]||u|| = \sqrt{(5)^{2} + (3)^{2} + (6)^{2} + (4)^{2}} = \sqrt{86} = 2\sqrt{43}[/tex]
The Euclidean distance:
[tex]E = \sqrt{(5 - (-8))^{2} + (3 - (-4))^{2} + (6 - 6)^{2} + (4 - 3)^{2}} = \sqrt{219}[/tex]
Cosine of the angle
[tex]u.v = (-8,-4,6,3).(5,3,6,4) = -40 -12 +36 +12 = -4[/tex]
[tex]\cos{\alpha} = \frac{u.v}{||u||||v||} = -\frac{4}{10\sqrt{215}}[/tex]
Since the cosine is negative, the angle is obtuse.
