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Unit Vectors quiz #1

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  • How do you express a vector, such as vector b, using unit vectors in the Cartesian coordinate system?
    To express a vector b using unit vectors in the Cartesian coordinate system, write it as the sum of its components along the x, y, and z axes multiplied by the corresponding unit vectors: b = b_x i + b_y j + b_z k, where b_x, b_y, and b_z are the scalar components of b in the x, y, and z directions, and i, j, and k are the unit vectors in those directions.
  • What is the magnitude of a unit vector by definition?
    A unit vector always has a magnitude of one. This property allows it to indicate direction only, not length.
  • In which directions do the unit vectors i, j, and k point in the Cartesian coordinate system?
    The unit vector i points in the positive x direction, j in the positive y direction, and k in the positive z direction. These directions correspond to the axes of the Cartesian system.
  • How does unit vector notation simplify the process of vector addition?
    Unit vector notation allows you to add vectors by simply adding their corresponding i, j, and k components. This eliminates the need for decomposing vectors into magnitudes and angles.
  • If a vector is written as 3i + 4j, what are its x and y components?
    The x component is 3 and the y component is 4. These values represent the number of units in the i and j directions, respectively.
  • What does a negative coefficient in front of a unit vector indicate about the direction of the vector component?
    A negative coefficient means the component points in the negative direction of that axis. For example, -i points in the negative x direction.
  • How can you graphically interpret the vector 4i + 2j?
    You move 4 units in the x direction and 2 units in the y direction from the origin. The resulting point gives the tip of the vector.
  • What is the resultant vector when adding 4i + 2j and -i + 2j using unit vector notation?
    The resultant vector is 3i + 4j. This is found by adding the i components (4 + -1) and the j components (2 + 2).
  • Why did physicists introduce the i, j, and k notation for vectors?
    Physicists introduced this notation to provide a standardized and clear way to represent vector components along each axis. It helps avoid confusion and streamlines vector calculations.
  • What does the vector 3i + 4j represent in terms of a right triangle?
    It represents a right triangle with legs of length 3 and 4 along the x and y axes, respectively. The hypotenuse of this triangle is the vector's magnitude.