Is there any difference between angle and direction in relation to 2D space? Can one of them help me in a way that the other can not? the only difference that I can see is that angle is single value and the other is a vector.
Thank you in advance.
Difference between angle and direction in two dimensional space?
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linear-algebra
trigonometry
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0Intuitively I like to think of the set of parallel vectors that "point the same way" as having the same direction; and think of an angle as a difference in direction. – 2017-02-13
2 Answers
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Depends on what exactly you mean by "direction".
One way to "describe" the word direction is by sayin that it's the "property" that $x$ shares with $\lambda x$ for all $\lambda>0$. In this case, you can "represent" the direction of a nonzero vector $x$ by the normalized vector $\frac{x}{\|x\|}$, and the set of all directions is the set $S^1$ (i.e., the unit circle).
This set is in a 1-to-1 correspondence with the set of all angles, as you can map any angle $\theta$ to the point $(\cos\theta, \sin\theta)$.
So, in this way, directions and angles, in $2D$, are two sides of the same coin -- if I tell you the angle between a vector and the positive $x$-axis, I give you the exact same information as I would if I told you what the normalized version of that vector is.
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From an axiomatic point of view one can say that angle is a numerical measurement of direction.
In a formal treatment of Euclid's axioms, for example, "direction" is defined first (perhaps one might say that "direction" and "ray" are equivalent), and "angle" is then introduced as a numerical measurement assigned to a pair of directions: the angle between two directions represents intuitively the amount one must rotate one direction to get to the other.
Next, when a Cartesian $x,y$-coordinate system is imposed on a bare Euclidean plane, each individual direction may be assigned an angle, equal to the angle between that direction and the direction of the positive $x$-axis.