Name: vector orientation with direction cosines
Uploaded: 2025-05-22T09:06:01.000Z
Duration: 1 min 29 s
Channel: JoVE

i、j、k 単位ベクトルを使用してデカルト ベクトル形式で表されるベクトル A を考えます。ベクトル A の大きさは、その成分の平方和の平方根として定義されます。
このベクトルの方向は、正の x、y、z 軸から測定された座標方向角度 (α、β、γ) によって定義されます。 
これらの角度は、Aをそれぞれの軸に投影することで決定できます。これらは、Aの方向余弦として知られています。
これらの方向余弦は、単位ベクトルを使用して簡単に取得できます。ベクトル A に沿った単位ベクトルを考えます。この単位ベクトルは、ベクトル A をその大きさで割ることにより、デカルト形式で表すことができます。
式は、単位ベクトルの i、j、k 成分が A の方向余弦を表すことを示しています。
方程式を二乗すると、コサインの方向間に有意な関係を定式化できます。
あるいは、Aの大きさと座標方向角がわかっている場合は、デカルトベクトル形式で表すことができます。

<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-45、46、47。</li>
</ol>

ベクトルの方向余弦

方向余弦は、ベクトルの座標軸に対する向きを表すのに役立つ概念であり、ベクトル解析の分野では重要な概念です。ベクトル A は、i、j、およびkの単位ベクトルを使用してデカルトベクトル形式で表されるものとします。ベクトル A の大きさは、その成分の二乗の和の平方根として定義されます。このベクトルの方向は、座標方向角α、β、およびγによって定義されます。これらの角度は、ベクトル A をそれぞれの軸に射影することによって求めることができ、これがベクトル A の方向余弦として知られています。

<img alt="Equation 1" src="/files/ftp_upload/14233/14233_Equation_1.svg" style="text-align: center; height: 30px;" />

<img alt="Equation 1" src="/files/ftp_upload/14233/14233_Equation_2.svg" style="text-align: center; height: 30px;" />

<img alt="Equation 1" src="/files/ftp_upload/14233/14233_Equation_3.svg" style="text-align: center; height: 30px;" />

方向余弦の式を二乗することで、重要な関係を表すことができます。この関係は、方向余弦の二乗の和が１に等しいことを示します。

<img alt="Equation 1" src="/files/ftp_upload/14233/14233_Equation_4.svg" style="text-align: center; height: 15px;" />

この式を使用すると、座標角のうち2つがわかっている場合、3つ目の角を求めることができます。方向余弦は、空間内の成分に基づいてベクトルの向きを表すのに役立ち、物理学、工学、コンピュータグラフィックスなど、さまざまな分野で重要な概念です。ベクトルの方向余弦を理解することにより、その向きと変位を容易に決定することができます。その結果、正確なモデルとシミュレーションの開発が可能となります。

方向余弦は、座標軸に対するベクトルの向きを記述するのに役立ち、ベクトルの分野では重要な概念です...

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

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 resulting values are the direction cosines of A.

A key relationship comes from squaring the equation for the direction cosines. The sum of the squares of the direction cosines equals one. If two of the coordinate angles are known, this relationship can be used to find the third angle.

Direction cosines are useful for describing vector orientation from its components in space. They matter in physics, engineering, and computer graphics. They also help determine displacement and support accurate models and simulations.

Vector Orientation with Direction Cosines

vector-orientation-with-direction-cosines

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...

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introduction to force

Force, Motion, and Mechanical Effects

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force classification

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two-dimensional force system

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two-dimensional force system: problem solving

two-dimensional-force-system-problem-solving

Two-Dimensional Forces: Free-Body Analysis

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scalar notation

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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

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

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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...

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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 direction cosines link a vector’s components to its angle with the x, y, and z axes, and how they help find orientation in space.

Vector Orientation with Direction Cosines | 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 Orientation with Direction Cosines

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