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Vector Composition & Decomposition quiz

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  • What three things are needed to fully describe a vector without a grid?
    A vector is described by its magnitude, direction (angle relative to the x-axis), and its components (legs of the triangle).
  • What is vector composition?
    Vector composition is when you are given the components of a vector and you calculate the magnitude and direction of the vector.
  • What is vector decomposition?
    Vector decomposition is when you are given the magnitude and direction of a vector and you calculate its components.
  • How do you calculate the magnitude of a vector from its components?
    Use the Pythagorean theorem: magnitude = sqrt((x component)^2 + (y component)^2).
  • How do you calculate the direction (angle) of a vector from its components?
    The direction is found using the inverse tangent (arctangent) of the y-component divided by the x-component: θ = arctan(y/x).
  • What equation is used to find the x-component of a vector given its magnitude and angle?
    The x-component is found using: x = magnitude × cos(angle relative to x-axis).
  • What equation is used to find the y-component of a vector given its magnitude and angle?
    The y-component is found using: y = magnitude × sin(angle relative to x-axis).
  • Why must the angle used in decomposition be relative to the x-axis?
    Because the decomposition equations (using cosine and sine) are based on the angle measured from the x-axis, not the y-axis.
  • If a vector has components of 3 (x) and 4 (y), what is its magnitude?
    The magnitude is 5, calculated as sqrt(3^2 + 4^2).
  • If a vector has components of 3 (x) and 4 (y), what is its direction relative to the x-axis?
    The direction is 53 degrees, found by arctan(4/3).
  • Given a vector with magnitude 5 and angle 53° relative to the x-axis, what are its x and y components?
    The x-component is 3 (5 × cos(53°)), and the y-component is 4 (5 × sin(53°)).
  • What is the process for finding the magnitude and direction if you are given the x and y components?
    Use the Pythagorean theorem for magnitude and arctangent for direction.
  • What is the process for finding the components if you are given the magnitude and direction?
    Multiply the magnitude by the cosine of the angle for the x-component and by the sine of the angle for the y-component.
  • If a vector has components 8 (x) and 6 (y), what is its magnitude and direction?
    The magnitude is 10 (sqrt(8^2 + 6^2)), and the direction is 37° (arctan(6/8)).
  • If a vector has magnitude 13 and angle 67.4°, what are its x and y components?
    The x-component is 5 (13 × cos(67.4°)), and the y-component is 12 (13 × sin(67.4°)).