Respuesta :
Given vector: v=<42,10>
unit vector :
v1=<42,10>/sqrt(42^2+10^2)
=<42,10>/(2 √ 466)
=<21/sqrt(466),5/sqrt(466)>
=<0.9728,0.2316> approx.
unit vector :
v1=<42,10>/sqrt(42^2+10^2)
=<42,10>/(2 √ 466)
=<21/sqrt(466),5/sqrt(466)>
=<0.9728,0.2316> approx.
The unit vector n in the direction of v is 0.9728i + 0.2316j and the angle that the vector v made with the positive x-axis and y-axis is 13.38° and 103.38°.
What is a unit vector?
A vector is a quantity that has both magnitude, as well as direction. A vector that has a magnitude of 1 is a unit vector. Any vector can become a unit vector by dividing it by the vector's magnitude.
For the given situation,
The vector v = (42)i + (10)j
The unit vector n in the direction of v is
[tex]n=\frac{v}{|v|}[/tex]
⇒ [tex]\frac{(42)i + (10)j }{\sqrt{42^{2} +10^{2} } }[/tex]
⇒ [tex]\frac{(42)i + (10)j }{\sqrt{1764 +100} }[/tex]
⇒ [tex]\frac{(42)i + (10)j }{\sqrt{1864} }[/tex]
⇒ [tex]\frac{(42)i + (10)j }{43.17}[/tex]
⇒ [tex]0.9728i+0.2316j[/tex]
The angles made by the vector v = (42)i + (10)j with the positive x- and y-axes is
[tex]tan \theta = \frac{y}{x}[/tex]
Here, (x,y) is (42,10)
The angle that the vector v made with the positive x-axis is
⇒ [tex]tan \theta = \frac{10}{42}[/tex] [∵ Pythagoras theorem]
⇒ [tex]\theta=tan^{-1}\frac{10}{42}[/tex]
⇒ [tex]\theta=tan^{-1}(0.2380)[/tex]
⇒ [tex]\theta=13.38[/tex]
Then, the angle that the vector v made with the positive y-axis is
⇒ [tex]\theta=13.38+90[/tex]
⇒ [tex]\theta=103.38[/tex]
Hence we can conclude that the unit vector n in the direction of v is 0.9728i + 0.2316j and the angle that the vector v made with the positive x-axis and y-axis is 13.38° and 103.38°.
Learn more about unit vector here
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