3.3 Vector Addition and Subtraction: Analytical Methods

Vector addition and subtraction are key skills in physics. These methods allow us to combine or compare quantities with both magnitude and direction, like forces or velocities. Understanding how to break vectors into components and recombine them is crucial for solving many physics problems.

The notes cover analytical and graphical methods for vector operations. They explain how to find vector components, add or subtract vectors, and determine the characteristics of resultant vectors. These techniques are essential for analyzing complex systems and predicting outcomes in physics scenarios.

Vector Addition and Subtraction

Principles of vector addition

Top images from around the web for Principles of vector addition

• 

  2.2 Coordinate Systems and Components of a Vector | University Physics Volume 1 View original

  Is this image relevant?

• 

  File:Vector addition.png - Wikimedia Commons View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods – Physics View original

  Is this image relevant?

• 

  2.2 Coordinate Systems and Components of a Vector | University Physics Volume 1 View original

  Is this image relevant?

• 

  File:Vector addition.png - Wikimedia Commons View original

  Is this image relevant?

1 of 3

Top images from around the web for Principles of vector addition

• 

  2.2 Coordinate Systems and Components of a Vector | University Physics Volume 1 View original

  Is this image relevant?

• 

  File:Vector addition.png - Wikimedia Commons View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods – Physics View original

  Is this image relevant?

• 

  2.2 Coordinate Systems and Components of a Vector | University Physics Volume 1 View original

  Is this image relevant?

• 

  File:Vector addition.png - Wikimedia Commons View original

  Is this image relevant?

1 of 3

• Vectors possess both magnitude and direction represented by arrows, with the length indicating magnitude and the arrowhead pointing in the direction of the vector (displacement, velocity)
• Vector addition and subtraction can be performed analytically using the component method breaks vectors into their horizontal (x) and vertical (y) components
• Components are added or subtracted separately, then combined to find the resultant vector (net force, total displacement)
• To add vectors analytically:
  1. Add the x-components of all vectors to find the resultant x-component
  2. Add the y-components of all vectors to find the resultant y-component
  3. Combine the resultant x and y components to determine the magnitude and direction of the resultant vector (resultant force, total velocity)
• To subtract vectors analytically:
  1. Subtract the x-component of the second vector from the x-component of the first vector
  2. Subtract the y-component of the second vector from the y-component of the first vector
  3. Combine the resultant x and y components to determine the magnitude and direction of the resultant vector (relative velocity, net acceleration)
• The parallelogram method is a graphical technique for vector addition, where vectors are arranged to form a parallelogram and the diagonal represents the sum

Analytical vector components

• To find the horizontal (x) component of a vector multiply the magnitude of the vector by the cosine of the angle it makes with the positive x-axis [object Object],[object Object], where $A$ is the magnitude of the vector and $\theta$ is the angle (eastward component of velocity)
• To find the vertical (y) component of a vector multiply the magnitude of the vector by the sine of the angle it makes with the positive x-axis [object Object],[object Object], where $A$ is the magnitude of the vector and $\theta$ is the angle (northward component of force)
• The angle $\theta$ is measured counterclockwise from the positive x-axis to the vector
• If the vector is in the second quadrant pointing up and to the left, the x-component will be negative (northwest wind)
• If the vector is in the fourth quadrant pointing down and to the right, the y-component will be negative (southeast acceleration)
• Orthogonal decomposition involves breaking a vector into perpendicular components, typically along the x and y axes

Resultant vector characteristics

• To find the magnitude of the resultant vector use the Pythagorean theorem to calculate the magnitude from the resultant x and y components [object Object],[object Object], where $R$ is the magnitude of the resultant vector, $R_x$ is the resultant x-component, and $R_y$ is the resultant y-component (total displacement from origin)
• To find the direction of the resultant vector calculate the angle $\theta$ that the resultant vector makes with the positive x-axis using the inverse tangent function $\theta = \tan^{-1} (\frac{R_y}{R_x})$, where $R_y$ is the resultant y-component and $R_x$ is the resultant x-component
• The angle $\theta$ will be in the quadrant determined by the signs of the resultant x and y components
  1. First quadrant with both positive: $0° \leq \theta < 90°$ (northeast resultant force)
  2. Second quadrant with x negative, y positive: $90° \leq \theta < 180°$ (northwest resultant velocity)
  3. Third quadrant with both negative: $180° \leq \theta < 270°$ (southwest net acceleration)
  4. Fourth quadrant with x positive, y negative: $270° \leq \theta < 360°$ (southeast total displacement)

Vector and Scalar Quantities

• A vector is a quantity with both magnitude and direction, existing in a vector space
• A scalar is a quantity with only magnitude, such as temperature or mass
• Unit vectors are vectors with a magnitude of 1, often used to indicate direction in a coordinate system