MCV4U – Grade 12 Calculus & Vectors – Geometric Vectors Test

Vectors

  • Vector: is a quantity that has direction and magnitude
  • Scalar: is a quantity that only has magnitude
  • True Bearing: is directed compass measurement, beginning at North and rotating clockwise.
  • Quadrant Bearing: is a compass measurement east or west of the North-South line
  • Equivalent Vectors: are equal in magnitude and direction.
    • Vectors can be translated anywhere on the same plane and still be equivalent
  • Oppose Vectors: are equal in magnitude, but opposite in direction
  • Vectors can be

 

Adding and Subtracting Vectors

  • Since vectors remain equivalent no matter where they are translated, they can be moved around to construct diagrams more convenient for solving.
  • Adding Vectors using Tail to head method using a triangle or tail to tail method using a parallelogram.
  • The resultant is a vector joining the head of the first vector to the tail of the last vector
  • If 2 vectors, a and b, are parallel in the same direction, |a + b| = |a| + |b| and in the same direction
  • If a and b have opposite directions and | a | > | b |, then | a + b | = | a | – | b | and a + b is in the same direction as a.
  • Subtract vectors by adding the opposite vector
  • Zero Vector: means there is no magnitude or direction. Addition of 2 opposite vectors
  • Vectors follow commutative, associative, and identity properties
    • They can be added in any order
    • Simplifying vector expressions is similar to simplifying integer expressions

 

Scalar Vector Multiplication

  • When a vector is multiplied by a scalar, the magnitude is multiplied by the scalar and the vectors are parallel. The directions remains unchanged if the scalar is positive, and becomes opposite if scalar is negative.
  • Multiplying vectors follow the distributive and associative rule
    • They can be expanded using FOIL
    • They can be multiplied in any order
  • Linear Combination of vectors can be formed by adding scalar multiples of 2 or more vectors.

 

Applications of Geometric Vectors

  • 2 Vectors that are perpendicular to each other and add together to give a vector v are called the rectangular vector components of v.
  • When solving resultants, you can use Vector operations, pythagorean theorem, or trigonometry.
  • Equilibriant Vector: is the opposite of the resultant