The angle between vector and is approximately radians, which is equivalent to approximately .
Step-by-step explanation:
The angle between two vectors can be found from the ratio between:
their dot products, and
the product of their lengths.
To be precise, if denotes the angle between and (assume that or equivalently ,) then:
.
<h3>Dot product of the two vectors</h3>
The first component of is and the first component of is also
The second component of is while the second component of is . The product of these two second components is .
The dot product of and will thus be:
.
<h3>Lengths of the two vectors</h3>
Apply the Pythagorean Theorem to both and :
.
.
<h3>Angle between the two vectors</h3>
Let represent the angle between and . Apply the formula to find the cosine of this angle:
.
Since is the angle between two vectors, its value should be between and ( and .) That is: and . Apply the arccosine function (the inverse of the cosine function) to find the value of :
The "Pythagorean relation" between trig functions can be used to find the sine.
<h3>Pythagorean relation</h3>
The relation between sine and cosine is the identity ...
sin(x)² +cos(x)² = 1
This can be solved for sin(x) in terms of cos(x):
sin(x) = √(1 -cos(x)²)
<h3>Application</h3>
For the present case, using the given cosine value, we find ...
sin(x) = √(1 -(√3/2)²) = √(1 -3/4) = √(1/4)
sin(x) = 1/2
__
<em>Additional comment</em>
The sine and cosine of an angle are the y and x coordinates (respectively) of the corresponding point on the unit circle. The right triangle with these legs will satisfy the Pythagorean theorem with ...
sin(x)² + cos(x)² = 1 . . . . . . where 1 is the hypotenuse (radius of unit circle)
A calculator can always be used to verify the result.