Name: vector components in 3d space
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 28 s
Channel: JoVE

ベクトル A の x と y の成分が単位ベクトル i と j で表されているとします。ここで、単位ベクトルの無次元の大きさは 1 です。
ベクトル成分の大きさは常に正の量であり、スカラーで表されるため、A はデカルトベクトルとして表すことができます。
ここでは、右手の長方形座標系が使用されます。右手の親指は正の z 軸を指し、指は正の x 軸から正の y 軸に向かって曲がります。
3 次元ベクトルは、i、j、および k 単位ベクトルを使用して直交デカルト座標で表すことができます。これらのベクトルの方向は、正または負の軸に応じて表されます。
ベクトルは、個々の成分のベクトル和として表され、その大きさは、その成分の二乗和の正の平方根として表されます。
ベクトル代数演算は、ベクトルをデカルト形式で表すことで簡略化されます。単位ベクトル表記を使用して、軸に沿って大きさと方向を分離します。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。ニュージャージー州ホーボーケン:ピアソンプレンティスホール。ページ番号-34、44、45。</li>
</ol>

デカルト座標表記

デカルト座標表記は、機械工学において、三次元空間でのベクトルを表現し、勾配、発散、カールなどのベクトル演算を行い、変位、速度、加速度、力などの物理量を表現するための貴重なツールです。デカルトベクトル表記を使用することにより、エンジニアは力学、運動学、流体力学など、様々な機械工学の分野で問題をより簡単に分析し、解決することができます。この表記は、ベクトルをx軸、y軸、およびz軸に沿った3つの成分で表現します。

例えば、ベクトルAが(3, -4, 5)の方向を指しているとします。この場合、デカルトベクトル表記を使用してAは以下のように表現されます：A = 3i - 4j + 5kです。ここで、i、j、kはそれぞれx軸、y軸、およびz軸に沿った単位ベクトルです。単位ベクトルは次のように定義されます：i = (1, 0, 0)、j = (0, 1, 0)、k = (0, 0, 1)。

デカルトベクトル表記は、加算、減算、スカラー倍などのさまざまなベクトル演算に使用することができます。例えば、ベクトルA = 3i - 4j + 5kおよびベクトルB = 2i + 7j - 3kが与えられた場合、以下のようにデカルトベクトル表記を使用してそれらを加算することができます：

<img alt="Equation 1" src="/files/ftp_upload/14232/14232_ Equation_1v2.svg" style="text-align: center; height: 13px;" />

また、以下のように減算することもできます：

<img alt="Equation 2" src="/files/ftp_upload/14232/14232_ Equation_2v2.svg" style="text-align: center; height: 13px;" />

デカルトベクトル表記は、3次元空間でベクトルを表現し、ベクトル操作を実行するための機械工学の貴重なツールです...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Force Vectors

力のベクトル

KEY TERMS AND CONCEPTS

主要用語と概念

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, acceleration, and force. It also helps engineers work with dynamics, kinematics, and fluid mechanics. Cartesian vector notation can also support vector operations such as addition, subtraction, and scalar multiplication.

The same notation is also used with gradient, divergence, and curl. These operations are important in mechanical engineering calculations. For example, two vectors such as A = 3i - 4j + 5k and B = 2i + 7j - 3k can be added or subtracted using their components.

Vector Components in 3D Space

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

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...

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

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 Cartesian vector notation uses i, j, and k to show 3D vectors, describe force and motion, and perform addition, subtraction, and scalar multiplication.

Vector Components in 3D Space | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Vector Components in 3D Space

Description

Transcript