Final answer:
A vector can be replaced by its X and Y components, which are at right angles, by using trigonometric functions, specifically the cosine and sine of the vector's angle, along with the Pythagorean theorem for calculating the vector's magnitude.
Step-by-step explanation:
A single vector can be replaced by two vectors in the X- and Y-directions by using trigonometric functions. These two vectors are at right angles to each other, forming a right triangle with the original vector. This process of breaking a vector into its X (horizontal) and Y (vertical) components is often referred to as resolving the vector into its components.
For a vector A with an angle θ, the X-component of the vector (Δx) is found by Ax = A cos θ, and the Y-component of the vector (Δy) is found by Ay = A sin θ. The use of the Pythagorean theorem allows us to calculate the magnitude of the resultant vector when these two vectors are at right angles to each other.