2.2 Vectors Addition and Scalar Multiplication in 2D

Addition of Vectors in 2D

• Vectors are represented in coordinate form: (x1, y1) and (x2, y2).

• The sum of two vectors:

  • Sum = (x1 + x2, y1 + y2).

• Geometric representation involves:

  • Drawing the vectors in the XY space.

  • X components: Place vector (x2, y2) tip-to-tail with vector (x1, y1) in the XY plane.

  • Y components: Similar addition for y-coordinates.

• Resulting vector (red vector) represents the sum of the two vectors.

Geometric Interpretation of Vector Addition

• Addition can be visualized through:

  • Adding components separately or placing vectors tip-to-tail.

• Addition of vectors is commutative:

  • (x1, y1) + (x2, y2) = (x2, y2) + (x1, y1).

Scalar Multiplication of Vectors

• Algebraic representation:

  • a * (x1, y1) = (a * x1, a * y1).

• Geometrical implications:

  • If a > 1, the vector length increases (stretched).

  • If a < 1, the vector length decreases (shrunk).

  • Example: a = 1.5 results in a vector 150% longer than the original length.

Behavior with Negative Scalars

• Scalar multiplication can result in negative values:

  • Example: a = -1.5 for vector (5, 6) results in (-7.5, -9).

• Negative scalars flip the vector's direction through multiplication of components by negative values.