ModelCourses/Calculus/Vectors/Vectors in Space

Vectors in Space

• Vector Algebra
  • The right-handed coordinate system, three axes, three coordinate planes and eight octants
    • Sketch a point in space.
    • Sketch a line that passes through a given point and is parallel to an axis.
    • Sketch a plane that contains a point and is parallel to a coordinate plane.
    • Sketch a plane that contains a point and is perpendicular to an axis.
    • Express a vector from Point A to Point B in vector notation.
    • Sketch a position vector.
  • Vector algebra: (1) scalar multiplication; (2) vector addition and subtraction
    • Compute and sketching a scalar times a vector and a sum (difference) of two vectors.
  • Triangle inequality

• Dot Product and Applications
  • Two definitions of dot product of two vectors
  • Angle of two vectors
    • Computing the dot product of two vectors
    • Computing the angle between two vectors
    • Determining if two vectors are parallel or orthogonal (perpendicular) when cosine of the angle is 1, -1, or 0
    • Determining if the angle of two vectors is acute, or obtuse when the dot product of two vectors is positive or negative
    • Given a vector u, create a vector that is parallel to u
    • Given a vector u, create a vector that is orthogonal to u
    • Given a vector u and an angle theta, create a vector v such that the angle of u and v is theta
  • Projection of vector u onto vector v
    • Work done by a force vector along a direction vector

• Cross Product and Applications
  • Computation of the cross product of two vectors
  • The cross product of vectors u and v is orthogonal (perpendicular) to u and v
    • Given two vectors u and v that are not parallel, find a vector which is orthogonal to both u and v
    • Given two vectors, determine a vector which is normal

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ModelCourses/Multivariate Calculus