A vector has horizontal and vertical components.
The value of [tex]\mathbf{\theta }[/tex] is 45 degrees, when the horizontal and vertical components are equal
Let the magnitude of the vector be H.
So, the horizontal (h) and the vertical components (v) of the vector would be:
[tex]\mathbf{v = Hcos(\theta)}[/tex] --- vertical
[tex]\mathbf{h = Hsin(\theta)}[/tex]
Equate both expressions
[tex]\mathbf{h = v}[/tex]
Substitute values for h and v
[tex]\mathbf{Hcos(\theta) = Hsin(\theta)}[/tex]
Divide both sides by H
[tex]\mathbf{cos(\theta) = sin(\theta)}[/tex]
When [tex]\mathbf{cos(\theta) = sin(\alpha)}[/tex], then
[tex]\mathbf{\theta + \alpha = 90^o}[/tex]
So, we have:
[tex]\mathbf{\theta + \theta = 90^o}[/tex]
[tex]\mathbf{2\theta = 90^o}[/tex]
Divide both sides by 2
[tex]\mathbf{\theta = 45^o}[/tex]
Hence, the value of [tex]\mathbf{\theta }[/tex] is 45 degrees
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