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🔋college physics i – introduction review

Perpendicular Components

Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025

Definition

Perpendicular components refer to the components of a vector that are at right angles, or perpendicular, to each other. These components are used to graphically represent and analyze the magnitude and direction of vectors in the context of vector addition and subtraction.

5 Must Know Facts For Your Next Test

  1. Perpendicular components are essential for graphically representing and analyzing the addition and subtraction of vectors.
  2. The perpendicular components of a vector can be used to determine the magnitude and direction of the resultant vector.
  3. Perpendicular components are often represented using the $x$- and $y$-axes, with the $x$-component being the horizontal component and the $y$-component being the vertical component.
  4. The Pythagorean theorem can be used to calculate the magnitude of a vector from its perpendicular components.
  5. The angle between a vector and its perpendicular components can be used to determine the direction of the vector.

Review Questions

  • Explain how perpendicular components are used to graphically represent and analyze vector addition.
    • Perpendicular components are essential for graphically representing and analyzing vector addition. By breaking down each vector into its $x$- and $y$-components, you can use the rules of vector addition to find the resultant vector. The $x$-components are added together, and the $y$-components are added together, to determine the magnitude and direction of the final vector. This graphical approach allows you to visualize the addition of vectors and understand how their perpendicular components contribute to the overall result.
  • Describe how the Pythagorean theorem can be used to calculate the magnitude of a vector from its perpendicular components.
    • The Pythagorean theorem can be used to calculate the magnitude of a vector from its perpendicular components. The theorem states that in a right triangle, the square of the length of the hypotenuse (the vector) is equal to the sum of the squares of the lengths of the other two sides (the perpendicular components). Mathematically, this can be expressed as $\text{magnitude} = \sqrt{x^2 + y^2}$, where $x$ and $y$ are the perpendicular components of the vector. This relationship allows you to determine the overall magnitude of a vector based on the values of its perpendicular components.
  • Analyze how the angle between a vector and its perpendicular components can be used to determine the direction of the vector.
    • The angle between a vector and its perpendicular components can be used to determine the direction of the vector. By considering the relative positions of the $x$- and $y$-components, you can infer the angle of the vector with respect to the positive $x$-axis. For example, if the $y$-component is positive and the $x$-component is negative, the vector would be in the second quadrant, with an angle between 90 and 180 degrees. Similarly, the other quadrants can be identified based on the signs of the perpendicular components, allowing you to determine the precise direction of the vector.