Before we learn the direction of a vector formula, let us recall what is a vector. A vector is a physical quantity having direction and magnitude both. The magnitude of a vector is its length whereas the direction of a vector is the angle made by it with the horizontal. Let us learn the direction of a vector formula along with a few solved examples.
What Is the Direction of a Vector Formula?
The direction of a vector formula is related to the slope of a line. We know that the slope of a line that passes through the origin and a point (x, y) is y/x. We also know that if θ is the angle made by this line, then its slope is tan θ, i.e., tan θ = y/x. Hence, θ = tan-1 (y/x). Thus, the direction of a vector (x, y) is found using the formula tan-1 (y/x) but while calculating this angle, the quadrant in which (x, y) lies also should be considered. So we find the direction of a vector using the following steps.
To find the direction of a vector (x, y):
Find α using α = tan-1 |y/x|.
Find the direction of the vector θ using the following rules depending on which quadrant (x, y) lies in:
Quadrant in which (x, y) lies
θ (in degrees)
180 - α
180 + α
360 - α
To find the direction of a vector whose endpoints are given by the position vectors \((x_1,y_1)\) and \((x_2,y_2)\), then to find its direction:
Find (x, y) using (x, y) = \((x_2-x_1,y_2-y_1)\)
Find α and θ just as explained earlier.
Let us see the applications of the direction of a vector formula in the following solved examples.
Breakdown tough concepts through simple visuals.
Math will no longer be a tough subject, especially when you understand the concepts through visualizations.