Operation with Vectors

Operations with Vectors

Constructing a Motion Diagram

  • Steps to Create a Motion Diagram

    1. Draw dots to represent the position of the object with respect to the observer at equal time intervals.

    2. Point velocity arrows in the direction of motion to indicate approximately how fast the object is moving, adjusting their lengths accordingly.

    3. Draw a velocity change arrow to indicate changes between adjacent position arrows.

  • Understanding motion requires knowledge of vector addition (e.g., V2 + V3 = V5) and subtraction (V3 - V2).

Understanding Vectors

  • Vector Representation

    • Vectors are represented by arrows; their length denotes magnitude and direction.

    • Each vector has a tail (starting point) and a head (ending point).

  • Examples from Figure 2.6

    • Arrows 1, 3 & 4 represent vector A.

    • Arrows 2, 5 & 7 represent vector B.

    • Arrows 6 & 8 are negative vectors (-A for vector 6 and -B for vector 8). Being negative means they have the same magnitude but opposite direction.

Rules of Vector Addition and Subtraction

  • Concepts from Figure 2.6b & 2.6c

    • In addition, vectors are combined to form a resultant vector.

    • In subtraction, vector B is taken away from vector A to form a new resultant vector.

Graphical Vector Addition

  • Adding Vectors K and L

    • Start with vector K, and move vector L so that its tail touches the head of K.

    • The resultant vector R is drawn from the tail of K to the head of L, represented as R = K + L.

    • When two vectors are parallel and point in the same direction, their magnitudes simply sum.

    • For antiparallel vectors, their magnitudes are subtracted, taking the absolute value.D

Graphical Vector Subtraction

  • Subtracting Vector P from M

    • To perform subtraction, first draw -P (the inverse of P).

    • The operation can be interpreted as adding M and -P, where Q = M - P = M + (-P).

  • Parallel and Antiparallel Vector Subtraction

    • Equivalent steps apply as in addition: align vector tails, with direction adjustments based on positivity or negativity.

Multiplying a Vector by a Scalar

  • Scalar Multiplication

    • Multiplying a vector by a scalar affects its magnitude without altering direction if a positive scalar is used, or reverses the direction if a negative scalar is applied.

    • The resultant vector is parallel or antiparallel to the original, with a magnitude equal to the product of the original vector’s magnitude and the absolute value of the scalar.