Study Notes on Vectors and Their Operations

Introduction to Vectors

• Vectors consist of two parts:

  • Location in space: Determined by a point without an arrow.
  • Direction and magnitude: Shown by the arrow's length and direction.

• Important note: For vectors that represent displacement, the vector's location is at the endpoint.

Vector Equality

• Two vectors are considered equal if:
  • They have the same length and same direction.
• Example: Vector A and vector A prime are equal if they meet the above criteria.

Negative Vectors

• The negative of a vector points in the opposite direction.
• In displacement, the starting point and endpoint serve as meaningful positions:
  • Displacement is the difference in distance between the starting and endpoint.
  • Negative displacement points in the opposite direction of the original displacement.
  • Similar concept applies to forces: the negative of a force points in the opposite direction.

Graphical Addition of Vectors

• Adding Two Vectors Graphically:

  • Place the vectors head to tail.
  • The vector sum (resultant) extends from the initial point of the first vector to the endpoint of the second vector.
  • Example: If vectors A and B are combined, the resultant vector C is the sum of A and B.
  • Commutative property: A + B = B + A.

• Parallelogram Method:

  • A vector can also be added by constructing a parallelogram where:
  • One side is vector A, another side is vector B, and final vector C goes from one corner to the opposite corner.

Magnitude of the Sum of Vectors

• The sum of vectors equals the sum of their magnitudes only when they are parallel.
• For antiparallel vectors (pointing in opposite directions): C equals the difference of their magnitudes.

Adding More than Two Vectors

• Vectors can be added in any quantity:
  • The addition of vectors is commutative.
  • Example: A + B + C = B + C + A = …
• When adding vectors graphically, it is easier to add them two at a time.

Subtracting Vectors

• Subtracting vectors is equivalent to adding the negative of a vector:
  • A - B = A + (-B).
• The negative flips the direction of the vector and behaves similarly during graphical representation.

Multiplying Vectors by Scalars

• Multiplying a vector by a scalar:
  • Does not change the vector's direction; only its magnitude is affected.
  • Example: If vector A is multiplied by a scalar of 2, the resulting vector is two times as long.
  • If multiplied by a negative scalar, the vector's direction reverses and its magnitude scales correspondingly.

Adding Perpendicular Vectors

• When adding two vectors at right angles, the Pythagorean theorem applies.
• Example Problem: A cross-country skier moves 1 km north and 2 km east.
  • To determine the resultant displacement, calculate the hypotenuse of the right triangle formed:
  • Hypotenuse (resultant vector) = $ext{Resultant} = ext{√(1 km}^2 + ext{2 km}^2)$.
• Use basic trigonometric functions to find angles and magnitudes in vector addition:
  • $A<em>x = A imes ext{cos}( heta)$ and $A</em>y = A imes ext{sin}( heta)$.

Conclusion

• Summary of key concepts:
  • Store key relationships for vector addition, subtraction, and multiplication by scalars.
  • Review the graphical methods for vector addition and subtraction for clarity in physical applications.