Final answer:
A two-dimensional vector's component shows displacement along the x- or y-axis and can be positive or negative. The component form of a vector is crucial in determining its direction and magnitude and performing vector addition.
Step-by-step explanation:
The student is asking about the component form of a two-dimensional vector. A component of a two-dimensional vector is a piece that points in either the x- or y-direction. When a vector is resolved into its scalar components, x and y, the signs of these components indicate the direction of the vector in relation to the coordinate axes. In other words, the x-component shows horizontal displacement (positive to the right and negative to the left of the origin), and the y-component shows vertical displacement (positive upward and negative downward from the origin).
For vectors located in different quadrants on the coordinate plane, their components will have different signs. For example, vectors in the first quadrant will have both components as positive, while vectors in the third quadrant will have both components as negative. The direction and magnitude of a vector can be found using the components and trigonometric functions.
Vector addition can be performed by adding corresponding components, that is by adding x-components together and y-components together. The result is a new vector represented by its own component form.