3.2 Vector Addition and Subtraction: Graphical Methods

Vector addition and subtraction are key skills for understanding forces and motion. These techniques allow us to combine or compare multiple vectors, which represent quantities with both magnitude and direction.

We'll explore methods like head-to-tail and parallelogram for adding vectors, as well as how to calculate resultant vectors. We'll also learn to break vectors into components and reconstruct them, essential for solving real-world physics problems.

Vector Addition and Subtraction

Vector addition and subtraction techniques

Top images from around the web for Vector addition and subtraction techniques

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

• 

  3.2 Vector Addition and Subtraction: Graphical Methods – College Physics: OpenStax View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

1 of 3

Top images from around the web for Vector addition and subtraction techniques

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

• 

  3.2 Vector Addition and Subtraction: Graphical Methods – College Physics: OpenStax View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

• 

  Vector Addition and Subtraction: Graphical Methods | Physics View original

  Is this image relevant?

1 of 3

• Head-to-tail method for vector addition connects the tail of the second vector to the head of the first vector, then draws the resultant vector from the tail of the first to the head of the last (displacement, velocity)
  • Reverse the direction of the vector being subtracted for vector subtraction, place the tail of the reversed vector at the head of the first vector, and draw the resultant from the tail of the first to the head of the reversed vector (force, acceleration)
• Parallelogram method for vector addition draws the two vectors with their tails connected, completes a parallelogram by drawing lines parallel to each vector, and the resultant vector is the diagonal from the connected tails to the opposite corner (force, velocity)
• Triangle method for vector addition arranges the vectors head-to-tail to form a triangle, and the resultant vector is the side that completes the triangle from the tail of the first vector to the head of the last (displacement, acceleration)
• These methods are performed within a coordinate system, which provides a reference frame for vector operations

Resultant vector calculations

• Measure the magnitude of the resultant vector using a ruler and convert the length to the appropriate scale based on the given vector magnitudes (5 cm = 10 N)
• Determine the direction of the resultant vector by measuring the angle between the resultant and a reference axis (positive x-axis) using a protractor (30° above the horizontal)
• Apply the Pythagorean theorem to calculate the magnitude of the resultant vector from its components $R = \sqrt{R_x^2 + R_y^2}$ (3 N horizontal, 4 N vertical, $R = \sqrt{3^2 + 4^2} = 5$ N)
• Use the arctangent function to find the angle between the resultant vector and the horizontal axis $\theta = \tan^{-1} \left(\frac{R_y}{R_x}\right)$ (3 N horizontal, 4 N vertical, $\theta = \tan^{-1} \left(\frac{4}{3}\right) \approx 53°$)
• Express the resultant using vector notation, which includes both magnitude and direction information

Vector component resolution

• Resolve a vector into its perpendicular components by determining the angle between the vector and the horizontal axis using a protractor (30° above the horizontal)
• Calculate the horizontal component using the cosine function $A_x = A \cos \theta$ (10 N at 30°, $A_x = 10 \cos 30° \approx 8.7$ N)
• Calculate the vertical component using the sine function $A_y = A \sin \theta$ (10 N at 30°, $A_y = 10 \sin 30° = 5$ N)
• Reconstruct a vector from its components by applying the Pythagorean theorem to find the magnitude $A = \sqrt{A_x^2 + A_y^2}$ (8.7 N horizontal, 5 N vertical, $A = \sqrt{8.7^2 + 5^2} \approx 10$ N)
• Use the arctangent function to find the angle between the reconstructed vector and the horizontal axis $\theta = \tan^{-1} \left(\frac{A_y}{A_x}\right)$ (8.7 N horizontal, 5 N vertical, $\theta = \tan^{-1} \left(\frac{5}{8.7}\right) \approx 30°$)

Vector and Scalar Quantities

• Vectors have both magnitude and direction, while a scalar only has magnitude
• Unit vectors are vectors with a magnitude of 1 and are used to indicate direction
• Equilibrium occurs when the vector sum of all forces acting on an object is zero, resulting in no net force