Solved Examples on Components of a Vector
Example 1: Consider a vector v with a magnitude of 5 units and an angle of 30 degrees with the positive x-axis. Find its x and y components.
Solution:
Using the trigonometric formulas for the components of a vector in two dimensions:
vx = |v| · cosθ
vy = |v| · sinθ
Given that ( |v| = 5 ) and θ = 30°
vx = 5 · cos30°, and
vy = 5 · sin30°
⇒ [Tex]v_x = 5 \cdot \frac{\sqrt{3}}{2} [/Tex], and
vy = 5 × 1/2
Simplifying further:
⇒ [Tex]v_x = \frac{5\sqrt{3}}{2} [/Tex], and
vy = 5/2
So, the x and y components of the vector (v) are [Tex]\frac{5\sqrt{3}}{2} [/Tex] and 5/2units, respectively.
Example 2: If a = (2,5) and b = (-1,3) , find the sum a+b in the component form.
Solution:
If a= (2, 5) and b = (-1, 3), find the sum a+b in component form.
The sum of two vectors is obtained by adding their corresponding components. So, for a+b:
a + b = (2 + (-1), 5 + 3)
⇒ a + b= (1, 8)
Therefore, a+b in component form is (1, 8).
Components of a Vector
Components of a Vector refer to its parts that contribute to its overall influence in a given coordinate system. Vectors, characterized by both magnitude and direction, can be effectively analyzed by breaking them down into components along specific axes. This breakdown typically occurs in two or three dimensions, with the components providing valuable insights into how the vector operates along each axis.
In this article, we will learn How to Find Components of Vector along with its definition, formula, and examples.
Table of Content
- What are the Components of the Vector?
- Formula of Components of a Vector
- How to Find the Components of a Vector?
- Components of a Vector Along b Vector
- Vector Addition with Components
- Types of Vector Components