Vector Addition and Mathematics

Graphical Vector Addition

  • The sum of two or more vectors is called the resultant vector:   C⃗=A⃗+B⃗\vec{C} = \vec{A} + \vec{B}

  • Vector addition is commutative, meaning vectors can be added in any order:   A⃗+B⃗=B⃗+A⃗\vec{A} + \vec{B} = \vec{B} + \vec{A}

  • Tip-to-Tail Rule: Slide the tail of the second vector to the tip of the first vector. The resultant vector points from the tail of the first vector to the tip of the second vector.

  • Parallelogram Rule: Place two vectors tail-to-tail. The resultant vector is the diagonal of the parallelogram formed by the two vectors.

Two vectors added using the tip-to-tail rule or parallelogram rule

  • For perpendicular vectors forming a right triangle, the magnitude and direction of the resultant vector are found using:   C=A2+B2C = \sqrt{A^2 + B^2}   θ=tan⁡−1(BA)\theta = \tan^{-1}\left(\frac{B}{A}\right)

Vector Subtraction and Scalar Multiplication

Working with vectors including scalar multiplication, negative vectors, and vector subtraction

  • Scalar Multiplication:

    • Multiplying vector A⃗\vec{A} by a positive scalar cc stretches or shrinks its magnitude to B=cAB = cA while maintaining the same direction.

    • Multiplying by a negative scalar changes the magnitude and reverses the vector's direction.

  • Negative Vectors: A vector −A⃗-\vec{A} has the exact same magnitude as A⃗\vec{A} but points in the opposite direction, satisfying:   A⃗+(−A⃗)=0⃗\vec{A} + (-\vec{A}) = \vec{0}

  • Vector Subtraction: Defined as adding the negative of the vector being subtracted:   A⃗−C⃗=A⃗+(−C⃗)\vec{A} - \vec{C} = \vec{A} + (-\vec{C})