Name: adding and multiplying vectors
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 20 s
Channel: JoVE

2차원 좌표계에서 두 개의 힘 벡터를 고려합니다.
이러한 벡터는 동시 벡터의 꼬리가 함께 결합되고 각 벡터에 평행한 선이 그려져 평행사변형을 형성하는 평행사변형 법칙을 사용하여 추가할 수 있습니다.
결과 벡터는 꼬리에서 선의 교차점까지 대각선을 그려서 얻을 수 있습니다.
마찬가지로, 삼각형 규칙을 사용하여 벡터를 추가할 수 있는데, 여기서 두 벡터는 삼각형의 두 변이 머리에서 꼬리로 연결되도록 배열되고 삼각형의 세 번째 변은 결과 벡터의 크기와 방향을 나타냅니다.
벡터 빼기는 한 벡터의 역을 다른 벡터에 더하는 것과 유사합니다. 결과적으로 두 벡터의 차이는 동일한 벡터 덧셈 규칙을 사용하여 구해집니다.
벡터에 스칼라를 곱하면 벡터의 크기는 스칼라 수량의 양에 따라 달라집니다.
스칼라가 양수이면 벡터의 방향이 동일하게 유지되고 음수이면 방향이 반대입니다.

<ol>
	<li>히벨러, R.C. (2016). Engineering Mechanics \u2012 정역학 및 동역학. 뉴저지 호보켄: 피어슨 프렌티스 홀. 18-19쪽</li>
</ol>

벡터 연산

벡터는 크기와 방향을 모두 갖는 물리량입니다. 벡터 연산에는 덧셈, 뺄셈, 스칼라 곱셈이 포함됩니다.

스칼라 값을 곱한 벡터를 스칼라 곱셈이라고 합니다. 얻은 결과는 크기가 다른 새로운 벡터가 생성됩니다. 스칼라가 양수이면 벡터의 방향은 동일하게 유지되지만 음수이면 벡터의 방향이 반전됩니다. 예를 들어, 질량과 속도의 곱은 운동량을 산출합니다. 여기서 스칼라 수량의 질량에 벡터 수량의 속도를 곱하면 벡터 수량의 운동량이 산출됩니다. 결과 운동량 벡터의 방향은 속도 벡터와 같은 방향입니다.

벡터 덧셈의 평행사변형 법칙을 사용하여 두 벡터를 기하학적으로 덧셈할 수 있습니다. 이 법칙을 사용하여 더하면 두 벡터가 평행사변형의 인접한 변으로 표시됩니다. 두 벡터의 꼬리를 원점에 놓고 각 벡터에 평행한 선을 그려 평행사변형을 만듭니다. 그런 다음 평행사변형의 대각선은 원점에서 평행사변형의 반대쪽 모서리까지 그려져 결과 벡터를 나타냅니다. 결과 벡터의 크기와 방향은 평행사변형 대각선의 길이와 각도를 측정하여 결정할 수 있습니다. 평행사변형 법칙은 두 벡터가 동일선상에 있는 특수한 시나리오에서 단순화될 수 있습니다. 즉, 동일한 작용선을 가집니다. 이 경우 두 벡터의 대수적 또는 스칼라 덧셈을 통해 결과 벡터를 얻을 수 있습니다.

두 개의 벡터를 그래픽적으로 더하는 또 다른 방법은 벡터 덧셈의 삼각형 법칙입니다. 벡터 덧셈의 삼각형 법칙은 두 개의 벡터가 삼각형의 두 변처럼 머리부터 꼬리까지 연결되어 배열되면 삼각형의 세 번째 변이 결과 벡터의 크기와 방향을 나타낸다는 것을 나타냅니다. 이 법칙을 사용하는 벡터 덧셈은 첫 번째 벡터를 원점에 배치하고 두 번째 벡터를 첫 번째 벡터와 머리부터 꼬리 방향으로 배치하는 작업이 포함됩니다. 두 번째 벡터의 원점에서 끝까지 삼각형의 세 번째 변은 결과 벡터를 나타냅니다.

벡터 뺄셈 과정에는 한 벡터의 역(음수)을 다른 벡터에 더하는 과정이 포함됩니다. 이는 벡터 덧셈의 규칙을 적용하여 두 벡터 간의 차이를 얻을 수 있음을 의미합니다.

벡터는 크기와 방향을 모두 가진 물리량입니다. 벡터 연산에는 더하기, 빼기, 스칼라 곱하기가 포함됩니다.
A ...

Education

교육

JoVE Core

Mechanical Engineering

기계공학

Unit 1

제1단원

Force Vectors

힘 벡터

KEY TERMS AND CONCEPTS

주요 용어 및 개념

Vector Operations

adding and multiplying vectors

Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses.

A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and velocity is the vector quantity. The result is momentum, and its direction matches the direction of the velocity vector.

Two vectors can be added geometrically with the parallelogram law. In this method, the two vectors are drawn as adjacent sides of a parallelogram with both tails at the origin. Lines are then drawn parallel to each vector to complete the shape.

The diagonal of the parallelogram, drawn from the origin to the opposite corner, shows the resultant vector. Its magnitude and direction can be found by measuring the diagonal's length and angles. When the two vectors are collinear, meaning they share the same line of action, the resultant can be found by algebraic or scalar addition.

Another way to add vectors is the triangle law of vector addition. Here, the vectors are arranged head to tail as two sides of a triangle. The third side of the triangle, from the origin to the end of the second vector, represents the resultant vector.

Vector subtraction uses the reverse, or negative, of one vector added to another vector. This lets the rules of vector addition be used to find the difference between two vectors.

Adding and Multiplying Vectors

scalar and vectors

Scalar vs Vector Quantities in Mechanics

Scalar and vector quantities are two main ways to describe physical quantities in mechanics. Common examples include force, speed, velocity, and work. Some of these are scalars, and others are vectors. A scalar is a quantity that has magnitude only. It does not need direction. Mass, area, and length are examples of scalar quantities. Scalars with the same units can be added or subtracted like ordinary numbers. A class that ends 10 min earlier than 50 min lasts 50 min minus 10 min, which is 40...

vector operations

Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses. A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and...

introduction to force

Force, Motion, and Mechanical Effects

Force changes the motion, shape, or orientation of an object. A flowing stream of water can push on a turbine vane and make it move in the direction of the flow. In mechanics, force is treated as a vector quantity, which means it has both magnitude and direction. It is measured in newtons, or N, in the SI unit system. Forces can act through direct contact or without contact. A push or pull is a contact force. Magnetic and gravitational forces are examples of non-contact interactions. For an...

force classification

Types of Forces in Physics

Forces in physics can be grouped by how they act and where they come from. These groups help describe motion, behavior, and equilibrium in the physical world. They are also useful in physics and engineering. One common way to classify forces is by whether they need contact. Contact forces act only when two objects touch. Frictional force, tension, and normal force are all contact forces that come from surface contact. Forces can also act without physical touch. These are called non-contact...

vector addition of forces

Resultant Force from Tugboats

Resultant force can be found with vector addition when two or more forces act on the same object. This idea helps explain how multiple pulls combine to create one net force. It is a key tool for understanding motion in physics. A clear example is a ship pulled by two small tugboats. The forces, F1 and F2, act at the same time but in different directions. The net force on the ship is the vector sum of those two forces. The parallelogram law gives a geometric way to add vectors. In this method,...

two-dimensional force system

Resolving Forces in a Plane

Resolving forces in a plane is a key part of two-dimensional mechanical analysis. A force vector can be split into x- and y-components, which show how the force acts in each direction. These components are written with the help of unit vectors, one for the x direction and one for the y direction. To find the components in a Cartesian coordinate system, first identify the angle theta between the force and the positive x-axis. If the line of action passes through the origin, basic trigonometry...

two-dimensional force system: problem solving

two-dimensional-force-system-problem-solving

Two-Dimensional Forces: Free-Body Analysis

Two-dimensional force analysis starts with a free-body diagram. A free-body diagram shows the object and every external force acting on it. It includes each force’s size, direction, and point of application. After the diagram is drawn, each force is broken into x- and y-components. This uses vector analysis and trigonometry, especially sine and cosine. The forces can then be written in Cartesian vector form, which makes them easier to work with. Next, the components in each direction are...

scalar notation

Vector Components for Force Calculations

Scalar notation helps simplify vector calculations by breaking a vector into separate x and y parts. When vectors are added or subtracted, each component can be handled on its own using scalar notation. This makes force problems easier to solve.

cartesian vector notation

Vector Components in 3D Space

Cartesian vector notation shows how to write vectors in three-dimensional space using x, y, and z components. It gives a clear way to represent direction and size for quantities used in mechanical engineering. The notation uses the unit vectors i, j, and k for the x, y, and z axes. A vector pointing in the direction (3, −4, 5) can be written as A = 3i - 4j + 5k. The unit vectors are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1). This form is useful for describing displacement, velocity,...

direction cosines of a vector

Vector Orientation with Direction Cosines

Direction cosines describe how a vector is oriented relative to the x, y, and z axes. For a vector A written in Cartesian form with i, j, and k unit vectors, this idea helps connect the vector’s components to its direction in space. The magnitude of vector A is the square root of the sum of the squares of its components. Its direction is described by the coordinate direction angles α, β, and γ with the x, y, and z axes. These angles can be found by projecting vector A onto each axis. The...

three-dimensional force system

Force Components in 3D Space

A three-dimensional force system describes forces acting in x, y, and z directions. In mechanical engineering, this framework helps students and engineers study how objects and structures behave in three-dimensional space. One common example is a force acting on a particle in space. That force can be split into three parts: the x-component, the y-component, and the z-component. Vector mathematics is used to calculate each component, and together they make up the full three-dimensional force...

three-dimensional force system:problem solving

three-dimensional-force-systemproblem-solving

Resolving Forces in 3D Coordinates

Three-dimensional force systems involve forces acting in three different directions at the same time. These problems are common in physics and engineering. They are used to find the resultant force, which helps predict or analyze how an object or structure will behave. The first step is to resolve each force into scalar components. A scalar component is the amount of force along one axis. Trigonometric functions and vector addition are used to break the force into parts and then combine those...

position vectors

Using Position Vectors to Locate Points

Position vectors are used to locate points in space and show the direction and distance between two points. In math and science, they help describe spatial relationships and make calculations easier. A position vector can connect one point to another point or show a point’s location relative to the origin of coordinates. For a point P(x, y, z) relative to the origin O, the position vector r runs from O to P. In Cartesian vector form, it is written as r = x i + y j + z k. Here, i, j, and k are...

force vector along a line

Finding a Cable Force from Two Points

A cable force can be found from the two points that define its line of action. In three-dimensional statics, this setup often appears with a pole and a cable anchored to the ground. The cable ends are labeled A and B, and their position vectors are written as P_A and P_B. The position vector from A to B is found using the triangle law of vector addition. First, subtract P_A from P_B to get P_AB. Then find the magnitude of P_AB by taking the square root of the sum of the squares of its...

dot product

Using Dot Product to Find Work and Angle

The dot product connects vectors, work, and direction in mathematics and physics. In engineering, it is found by multiplying the magnitudes of two vectors and the cosine of the angle between them. It is shown with a dot symbol between the vectors. A real-world example is a vehicle pulling an object along the ground with a rope. If the rope is angled above the horizontal axis, the work done can be calculated with the dot product of the applied force and the object’s displacement. This makes the...

dot product: problem solving

Using the Dot Product to Solve Vector Questions

The dot product helps solve vector questions by combining vector size and the cosine of the angle between them. It is useful when the angle is measured anti-clockwise and when a problem asks for a vector relationship in component form. A good first step is to identify the question being asked. Then list the given vectors and write them in Cartesian form, also called component form. This makes the data easier to use in a calculation. Next, decide whether the dot product is the right operation.

cross product

Cross Product Direction and Magnitude

The cross product is a vector operation in vector algebra that creates a third vector from two input vectors. Unlike the scalar product, it does not give a single number. The result is a vector perpendicular to both original vectors. The size, or magnitude, of the cross product depends on the magnitudes of both vectors and the sine of the angle between them. A larger angle between the vectors gives a larger cross product magnitude. The direction is found with the right-hand rule. If the...

key terms and concepts

Learn how vector addition, subtraction, and scalar multiplication work using parallelogram and triangle laws, plus a momentum example for high school science.

Adding and Multiplying Vectors | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Adding and Multiplying Vectors

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