The y-component of a vector refers to the vertical or up-and-down portion of the vector when it is represented graphically. It describes the magnitude of the vector's projection onto the y-axis, which is one of the two perpendicular axes used to plot vectors in a two-dimensional coordinate system.
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The y-component of a vector is one of the two key pieces of information needed to fully describe the vector's magnitude and direction.
The y-component is determined by drawing a perpendicular line from the vector's endpoint to the y-axis, and measuring the length of this line segment.
When adding or subtracting vectors graphically, the y-components of the vectors are combined to find the y-component of the resultant vector.
The y-component is crucial for determining the overall direction of a vector, as it works in conjunction with the x-component to establish the vector's angle relative to the coordinate axes.
Understanding the y-component is essential for visualizing and manipulating vectors in two-dimensional space, which is a fundamental skill in physics and mathematics.
Review Questions
Explain how the y-component of a vector is determined and what information it provides about the vector.
The y-component of a vector is determined by drawing a perpendicular line from the vector's endpoint to the y-axis and measuring the length of this line segment. The y-component represents the vertical or up-and-down portion of the vector, and provides information about the vector's magnitude and direction in the y-dimension. Knowing the y-component, along with the x-component, allows you to fully describe the vector's magnitude and angle relative to the coordinate axes.
Describe the role of the y-component in the graphical addition and subtraction of vectors.
When adding or subtracting vectors graphically, the y-components of the individual vectors are combined to determine the y-component of the resultant vector. This is done by aligning the vectors head-to-tail and then measuring the length of the line segment that extends vertically from the resultant vector's endpoint to the y-axis. The y-component of the resultant vector, along with its x-component, defines the overall magnitude and direction of the combined effect of the original vectors.
Analyze how the y-component of a vector, in conjunction with the x-component, is used to establish the vector's angle relative to the coordinate axes.
$$\tan\theta = \frac{y\text{-component}}{x\text{-component}}$$ The y-component and x-component of a vector work together to determine the vector's angle $$\theta$$ relative to the positive x-axis. By using the trigonometric tangent function, you can calculate the angle of the vector based on the ratio of its y-component to its x-component. This relationship is fundamental for visualizing and analyzing the direction of vectors in a two-dimensional coordinate system, which is essential for understanding vector addition and subtraction.
The x-component of a vector refers to the horizontal or left-to-right portion of the vector when it is represented graphically, describing the magnitude of the vector's projection onto the x-axis.