Direction of a vector formula
Direction of the vector formula calculates the angle formed by the vector along with the x-axis, i.e., the horizontal axis. We consider the direction concerning the vector angle’s counterclockwise rotation towards the east.
What is the direction of a vector formula?
The direction of any vector is simply its orientation with the horizontal axis. E.g. Velocity provides us with the magnitude demonstrated by an object when it is moving. Also, it tells us the direction of movement. Therefore velocity has to be a vector quantity.
Likewise, the force vector indicates the direction of an active force. The symbol a🡪 denotes a vector’s direction. It is expressed as a🡪 = ।a।â, here, ।a। is the vector’s magnitude while â is the unit vector that gives you the direction of this particular vector (a🡪).
Direction of the vector formula and the slope of line are interrelated. It is known that a slope is expressed as y/x when it goes through a point (x,y). Let us assume θ as the angle formed due to this line. Then the slope tan θ = y/x or, θ = tan-1y/x.
So, we need to determine tan-1y/x in order to get the vector’s direction. Also, you must consider the quadrant where the point (x,y) lies.
To get the vector’s direction whose endpoints are provided by two position vectors (x1,y1) & (x2,y2), we follow these steps:
The vector (x,y) needs to be determined using this formula:
(x,y) = (x2 –x1, y2 – y1)
In the next step, determine the values of θ and α. To know how it is done, read the following table:
Quadrant where (x,y) exists | θ (in degrees) |
1 | α |
2 | 180° – α |
3 | 180° + α |
4 | 360° – α |
Solved Examples
Find the vector’s direction whose initial point A = (1,3) and B = (4,9).
Solution: The vector AB’s coordinates are (x,y).
(x,y) = (x2 –x1, y2 – y1) = (4-1, 9-3) = (3,6)
Implementing the direction of a vector formula, θ = tan-1 (y/x) or tan-1(6/3) = 89.090°
Ans: Direction of AB is 89.090° as we can see the point (x,y) remains in the first quadrant.
Find the direction of (1, -√3) using the direction of the vector formula.
Solution: Provided, (x,y) = (1, -√3)
We know, α = tan-1(y/x)
or, α = tan-1 (-√3/1)
or, α = tan-1(-√3)
∴ α =- 60°
From the problem we know that (x,y) lies in the fourth quadrant as y coordinate is negative and x is positive. Therefore, the direction is: θ = 360° – 60° = 300°
Ans: 300°