Glossary

3.2 Vector Addition and Subtraction: Graphical Methods

3 min read•june 18, 2024

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 and .

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

Vector Addition and Subtraction

Vector addition and subtraction techniques

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  • for vector addition connects the tail of the second vector to the head of the first vector, then draws the from the tail of the first to the head of the last (displacement, velocity)
    • Reverse the direction of the vector being subtracted for , 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)
  • for vector addition draws the two vectors with their tails connected, completes a parallelogram by drawing lines parallel to each vector, and the is the diagonal from the connected tails to the opposite corner (force, velocity)
  • 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 , which provides a reference frame for vector operations

Resultant vector calculations

  • Measure the 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 to calculate the magnitude of the resultant vector from its components R=Rx2+Ry2R = \sqrt{R_x^2 + R_y^2} (3 N horizontal, 4 N vertical, R=32+42=5R = \sqrt{3^2 + 4^2} = 5 N)
  • Use the function to find the angle between the resultant vector and the horizontal axis θ=tan⁡−1(RyRx)\theta = \tan^{-1} \left(\frac{R_y}{R_x}\right) (3 N horizontal, 4 N vertical, θ=tan⁡−1(43)≈53°\theta = \tan^{-1} \left(\frac{4}{3}\right) \approx 53°)
  • Express the resultant using , which includes both magnitude and direction information

Vector component resolution

  • Resolve a vector into its by determining the angle between the vector and the horizontal axis using a protractor (30° above the horizontal)
  • Calculate the using the Ax=Acos⁡θA_x = A \cos \theta (10 N at 30°, Ax=10cos⁡30°≈8.7A_x = 10 \cos 30° \approx 8.7 N)
  • Calculate the using the Ay=Asin⁡θA_y = A \sin \theta (10 N at 30°, Ay=10sin⁡30°=5A_y = 10 \sin 30° = 5 N)
  • Reconstruct a vector from its components by applying the Pythagorean theorem to find the magnitude A=Ax2+Ay2A = \sqrt{A_x^2 + A_y^2} (8.7 N horizontal, 5 N vertical, A=8.72+52≈10A = \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 θ=tan⁡−1(AyAx)\theta = \tan^{-1} \left(\frac{A_y}{A_x}\right) (8.7 N horizontal, 5 N vertical, θ=tan⁡−1(58.7)≈30°\theta = \tan^{-1} \left(\frac{5}{8.7}\right) \approx 30°)

Vector and Scalar Quantities

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

Key Terms to Review (29)

Arctangent: The arctangent, denoted as tan^-1 or arctan, is the inverse trigonometric function of the tangent function. It is used to find the angle whose tangent is a given value, providing the angle in radians or degrees.
Commutative: Commutative property states that the order of addition or multiplication does not affect the result. In physics, it often applies to vector addition where the sum remains the same regardless of the order of vectors.
Components: Components are the individual parts of a vector that show its influence in different directions, typically along the x and y axes. They are essential for breaking down vectors to simplify analysis and calculations.
Coordinate System: A coordinate system is a mathematical framework used to uniquely identify the position of a point or an object in space. It provides a systematic way to describe the location of an entity relative to a defined origin and a set of reference axes.
Cosine function: The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It plays a vital role in understanding how angles relate to distances and directions, particularly when analyzing vectors. In graphical methods, the cosine function is essential for breaking down vectors into their components, aiding in both vector addition and subtraction.
Direction: Direction specifies the orientation of a vector in space and is typically described using angles or coordinates. It is essential in understanding how vectors operate within two-dimensional kinematics.
Displacement Vector: A displacement vector is a directed line segment that represents the change in position of an object. It describes the distance and direction an object has moved from its initial location to its final location.
Equilibrium: Equilibrium is a state of balance or stability, where the forces acting on a system are in a state of balance, and the system remains at rest or in a constant state of motion. This concept is fundamental in various areas of physics, including mechanics, thermodynamics, and electromagnetism.
Head-to-Tail Method: The head-to-tail method is a graphical technique used to add or subtract vectors by aligning the head of one vector with the tail of another. This method allows for the visual representation and determination of the resultant vector through the geometric addition or subtraction of the individual vectors.
Horizontal Component: The horizontal component of a vector or quantity is the projection of that vector or quantity onto the horizontal axis. It represents the portion of the vector or quantity that is parallel to the ground or a horizontal surface.
Magnitude: Magnitude is the size or length of a vector. It represents the distance from the origin to the point in a coordinate system.
Magnitude: Magnitude is a measure of the size or scale of a quantity, representing its absolute or relative value. It is a fundamental concept in physics, particularly in the context of vectors, forces, and displacement.
Parallelogram Method: The parallelogram method is a graphical technique used to add or subtract vectors by representing them as the sides of a parallelogram. It allows for the determination of the resultant vector's magnitude and direction through the construction of a parallelogram.
Perpendicular Components: Perpendicular components refer to the components of a vector that are at right angles, or perpendicular, to each other. These components are used to graphically represent and analyze the magnitude and direction of vectors in the context of vector addition and subtraction.
Pythagorean Theorem: The Pythagorean Theorem is a fundamental relationship in geometry that describes the connection between the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Resultant: The resultant is the vector that represents the sum of two or more vectors. It is often found using graphical methods such as the parallelogram or triangle method.
Resultant vector: A resultant vector is the single vector that has the same effect as two or more vectors added together. It represents the combined magnitude and direction of these vectors.
Resultant Vector: The resultant vector is the single vector that represents the combined effect of two or more vectors acting on an object. It is the vector sum of all the individual vectors, capturing the net displacement, force, or quantity being measured.
Scalar: A scalar is a physical quantity that has only magnitude and no direction. Examples include mass, temperature, and electric potential.
Scalar: A scalar is a physical quantity that has only a magnitude, or numerical value, and no direction. Scalars are contrasted with vectors, which have both a magnitude and a direction. Scalars are commonly used in physics to describe various physical properties and quantities.
Sine Function: The sine function is a periodic function that describes the y-coordinate of a point moving along a unit circle. It is one of the fundamental trigonometric functions and is widely used in various fields, including physics, engineering, and mathematics.
Tail: The tail of a vector is the starting point or origin of the vector. It is crucial in graphical methods for vector addition and subtraction.
Triangle Method: The triangle method is a graphical technique used to add or subtract vectors by constructing a triangle. This method allows for the visualization and determination of the resultant vector from the given vectors in a two-dimensional plane.
Unit Vector: A unit vector is a dimensionless vector with a magnitude of 1 that points in a specific direction. It is used to represent the direction of a vector without considering its magnitude.
Vector: A vector is a quantity that has both magnitude and direction. It is typically represented by an arrow where the length represents the magnitude and the arrowhead points in the direction.
Vector Component: A vector component is the projection of a vector onto a specific coordinate axis or direction. It represents the magnitude of the vector in that particular dimension or direction, allowing for the decomposition of a vector into its constituent parts along different axes.
Vector Notation: Vector notation is a mathematical representation of a vector, which is a quantity that has both magnitude and direction. It is used to describe and manipulate the properties of vectors in various applications, including physics, engineering, and mathematics.
Vector Subtraction: Vector subtraction is the process of finding the difference between two vectors by subtracting their corresponding components. It is a fundamental operation in vector mathematics that allows for the manipulation and analysis of vector quantities, which are essential in various fields of physics, engineering, and mathematics.
Vertical Component: The vertical component of a vector or quantity refers to the portion or magnitude of that vector or quantity that is oriented in the vertical or up-down direction. It represents the y-coordinate or height-related aspect of a vector or motion, independent of the horizontal or left-right direction.
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