Vectors Revision Year 1

• What is a scalar quantity defined by?


  A scalar quantity is defined completely by its magnitude.
• Give an example of a scalar quantity.


  The density of an object is an example of a scalar quantity.
• How does the value of a scalar quantity change with the coordinate system?


  The value of a scalar quantity is independent of the coordinate system used.
• What happens to the frequency of harmonic oscillation when the frame of reference moves?


  The frequency varies if the frame of reference moves with respect to the source of the oscillations.
• What are vectors defined by?


  Vectors are defined by both magnitude and direction.
• What are typical examples of vectors?

  Typical examples of vectors include velocity, acceleration, and force.
• How is a general vector represented in mathematics?


  A vector is a set of components that transform like a displacement when the coordinate system changes.
• How is a vector graphically represented?


  A vector is represented by an arrow, where the length indicates its magnitude.
• How do we denote vectors in notation?


  Vectors are denoted by bold font, underlining, or placing an arrow above the symbol.
• What notation is used for a displacement vector?


  A displacement vector is indicated by capital letters corresponding to each point.
• How is the magnitude of a vector indicated?


  The magnitude of a vector is indicated by absolute value brackets.
• What is a unit vector and how is it denoted?


  A unit vector indicates the direction of a vector and is denoted with a "hat" above the symbol.
• How can any vector be expressed in terms of its magnitude and direction?


  Any vector can be written as a product of its magnitude and a unitary directional vector.
• What is a vector field?


  • A vector field assigns a vector to each point in a subset of space.
  • It is useful for representing quantities that vary continuously in space and/or time.
• What is the purpose of the website https://earth.nullschool.net?


  The website allows exploration of vector fields of global Earth surface winds in real time.
• What do the basis vectors in three dimensions represent?


  The basis vectors represent three orthogonal directions in space.
• What are the common symbols for the basis vectors in Cartesian coordinates?


  The common symbols are ˆi, ˆj, and ˆk.
• What is the significance of orthonormal basis vectors?


  Orthonormal basis vectors have unit length and are orthogonal to each other.
• How can an arbitrary vector be represented in Cartesian coordinates?


  An arbitrary vector can be represented as \( v = (v_x, v_y, v_z) = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} \).
• What is the formula for the magnitude of a vector in three dimensions?


  The magnitude of a vector is given by \( |v| = \sqrt{v_x^2 + v_y^2 + v_z^2} \).
• How is the directional unit vector obtained?


  The directional unit vector is obtained by dividing the vector by its magnitude.
• What are direction cosines of a vector?


  Direction cosines are the cosines of the angles that the vector subtends with the basis directions.
• What is the formula for the direction cosine with respect to the x-axis?


  The direction cosine with respect to the x-axis is given by \( \cos(\alpha) = \frac{v_x}{\sqrt{v_x^2 + v_y^2 + v_z^2}} \).
• How do the directions in the Cartesian coordinate system behave?


  The directions in the Cartesian coordinate system are fixed for all points in space.
• How do basis vectors in polar coordinates differ from those in Cartesian coordinates?


  In polar coordinates, the basis vectors change their directions at different points in space.
• What does a position vector connect in a coordinate system?


  A position vector connects the origin of a coordinate system and a specific point in space.
• How is the position vector expressed in Cartesian coordinates?


  The position vector is expressed as \( r(P) = (x, y, z) = x \hat{i} + y \hat{j} + z \hat{k} \).
• How does the position vector change with time for an object in motion?


  The position vector changes with time as \( r(t) = x(t) \hat{i} + y(t) \hat{j} + z(t) \hat{k} \).
• How is the velocity vector obtained from the position vector?


  The velocity vector is obtained by taking the derivative of the position vector with respect to time.
• What is the formula for the velocity vector in terms of the position vector?


  The velocity vector is given by \( v(t) = \frac{dr(t)}{dt} = \frac{dx(t)}{dt} \hat{i} + \frac{dy(t)}{dt} \hat{j} + \frac{dz(t)}{dt} \hat{k} \).
• How is the acceleration vector related to the velocity vector?


  The acceleration vector is the second derivative of the position vector with respect to time.
• What is the formula for the acceleration vector?


  The acceleration vector is given by \( a(t) = \frac{d^2r(t)}{dt^2} = \frac{d^2x(t)}{dt^2} \hat{i} + \frac{d^2y(t)}{dt^2} \hat{j} + \frac{d^2z(t)}{dt^2} \hat{k} \).
• How can the trajectory of an object falling under gravity be expressed?


  The trajectory can be expressed in vector form as \( r(t) = r_0 + v_0 t - \frac{1}{2} g t^2 \).
• What are the component equations for the trajectory of an object falling under gravity?


  The component equations are \( x(t) = v_{0,x} t \) and \( y(t) = h_0 + v_{0,y} t - \frac{1}{2} g t^2 \).
• What does the equation \( y(x) = h_0 + \left(\frac{v_{0,y}}{v_{0,x}}\right)x - \left(\frac{g}{2v_{0,x}^2}\right)x^2 \) represent?


  This equation represents the vertical distance as a quadratic function of the horizontal distance, forming a parabola.
• How can vectors be represented as matrices?


  • Vectors can be represented as matrices with one row or one column.
  • This allows for expressing vectors and some operations in a concise form of matrix operations.
• How is a position vector expressed in terms of its components and basis vectors?


  A position vector can be expressed as \( r = \sum_{i=x,y,z} v_i \hat{e}_i \).
• What is the formula for the components of a position vector in a rotated coordinate system?


  The components are given by \( x' = r \cos(\beta - \alpha) \) and \( y' = r \sin(\beta - \alpha) \).
• How can the components of a vector in a rotated coordinate system be rewritten using trigonometric identities?


  The components can be rewritten as \( x' = x \cos(\alpha) + y \sin(\alpha) \) and \( y' = -x \sin(\alpha) \).
• What is the relationship between the original and rotated coordinates of a vector?


  The rotated coordinates are related to the original coordinates through trigonometric functions of the angle of rotation.