About The direction of a vector
The vector's direction is the angle formed by a vector with the horizontal axis or the X-axis. The counter-clockwise rotation of the angle of a vector about its tail due east determines its direction. A vector having a 45-degree direction, for example, is a vector that has been rotated 45 degrees counter-clockwise relative to due east. Another way to define a vector's direction is as an angle of rotation of the vector's tail from east to west, north to south. For example, if a vector's direction is 60 degrees north of the west, it means that the vector which is pointing towards the west has been rotated 60 degrees to the north.
The direction of a vector can be defined as the direction in which it acts. First, let's look at the formula for determining the direction of a vector and how to find the direction of a vector in different quadrants and some solved cases.
What does the direction of a vector mean?
The orientation of a vector is the angle it makes with the x-axis, which is its direction. A vector is created by drawing a line with an arrow at one end and a fixed point at the other. The vector's direction is determined by the direction in which the arrowhead is pointed. Velocity, for example, is a vector. It indicates the quantity of the object's movement as well as the direction in which the object is travelling. Similarly, the force vector determines the direction in which a force is exerted. A vector's direction is indicated by a? = |a|a?
Here, |a| is the magnitude of the vector and a? is the unit vector used to denote the direction of vector a?
The direction of a vector formula
The direction of a vector formula is closely related to the slope of a line. As we know that the slope of a line passing through the origin from a point (x, y) is y/x. If θ is the angle made by the line with the axis, then its slope is tan θ.
Thus, tan θ = y/x.
Hence, θ = tan-1(y/x).
The direction of a vector (x, y) can be calculated by using the above-given formula tan-1(y/x), the quadrant in which (x, y) lies also should be considered while calculating this angle.
Steps to find the direction of a given vector (x, y):
- Find the angle α using the formula α = tan-1|y/x|.
- Find the direction of the vector by calculating θ using the following given rules:
If we add or subtract zero from any number we will get the same number.
a + 0 = a ; 0 + a = a
| Quadrant in which (x, y) lies | θ (in degrees) |
| 1 | α |
| 2 | 180° - α |
| 3 | 180° + α |
| 4 | 360° - α |
To find the direction of a vector If endpoints are given in the form of position vectors (x1, y1) and (x2, y2), then the direction of the vector can be found as:
- Find the new vector (x, y) using the formula (x, y) = (x2 - x1, y2 - y1)
- Find α and θ, just as explained in previous steps. Check out the List of Maths Formulas.
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