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 between two vectors
    • Compute the dot product of two vectors.
    • Compute the angle between two vectors.
    • Determine if two vectors are parallel or orthogonal (perpendicular) when the cosine of the angle between these two vector is 1, -1, or 0.
    • Determine if the angle between two vectors is acute or obtuse when the dot product of these two vectors is positive or negative.
    • Create a vector v that is parallel to a given vector.
    • Create a vector v that is orthogonal to a given vector.
    • Given a vector u and an angle theta, create a vector v such that the angle between u and v is theta.
  • Projection and component of vector u onto vector v
    • Compute the work done by a force vector along a direction vector.
    • Compute the distance from a given point to a given line.
    • Compute the distance between two planes.

• Cross Product and Applications
  • Definition of the cross product of two vectors in space
  • The cross product of vectors u and v is orthogonal (perpendicular) to u and v and satisfies the right-handed rule.
    • Given two vectors u and v that are not parallel, find a vector which is orthogonal to both u and v.
    • Compute the area of the parallelogram whose two sides are formed by two given vectors.
    • Compute the volume of the parallelepiped whose three sides are formed by three given vectors.

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ModelCourses/Multivariate Calculus