Respuesta :
solution:
we know that ,
u.v = ΙuΙ ΙvΙcosθ
here,
θ =60° (since the given triangle is equilateral triangle)
u.v = ΙuΙ ΙvΙcos60°
= 1 x 1 x 1/2
u.v = 1/2
now, u.w = ΙuΙ ΙwΙcosθ
= ΙuΙ x cos(60x2)
u.w = -1/2
The scalar values are [tex]\vec u \,\bullet\, \vec v = \frac{1}{2}[/tex] and [tex]\vec u \,\bullet \, \vec w = \frac{1}{2}[/tex], respectively.
Vectorially speaking, we should use the definition of dot product to determine the scalar numbers, that is to say:
[tex]\vec u \,\bullet\,\vec v = \|\vec u\|\cdot \|\vec v\|\cdot \cos \alpha[/tex] (1)
[tex]\vec u\,\bullet \,\vec w = \|\vec u\|\cdot \|\vec w \|\cdot \cos (360^{\circ}-\alpha)[/tex] (2)
Where:
- [tex]\vec u[/tex], [tex]\vec v[/tex], [tex]\vec w[/tex] - Unit vectors.
- [tex]\alpha[/tex] - Internal angle of the equilateral triangle.
Please notice that equilateral triangles are triangles whose sides have the same length and all internal angles have the same measure. In addition, the sum of the measures of the internal angles in triangles equals 180°. Hence, the measure of each internal angle in equilateral triangles equals 60°.
If we know that [tex]\|\vec u\| = \|\vec v\| = \| \vec w\| = 1[/tex] and [tex]\alpha = 60^{\circ}[/tex], then the values of the scalar numbers are, respectively:
[tex]\vec u\,\bullet\,\vec v = (1)\cdot (1)\cdot \cos 60^{\circ}[/tex]
[tex]\vec u \,\bullet\, \vec v = \frac{1}{2}[/tex]
[tex]\vec u\,\bullet\,\vec w = (1)\cdot (1)\cdot \cos 300^{\circ}[/tex]
[tex]\vec u \,\bullet \, \vec w = \frac{1}{2}[/tex]
The scalar values are [tex]\vec u \,\bullet\, \vec v = \frac{1}{2}[/tex] and [tex]\vec u \,\bullet \, \vec w = \frac{1}{2}[/tex], respectively.
We kindly invite to check this question on dot product: https://brainly.com/question/16537974
