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

2 次元座標系の 2 つの力ベクトルについて考えます。
これらのベクトルは、平行四辺形の法則を使用して追加でき、並行ベクトルの裾が結合され、各ベクトルに平行な線が引かれて平行四辺形が形成されます。
結果として得られるベクトルは、尾部から線の交点まで対角線を引くことによって取得できます。
同様に、ベクトルは、2 つのベクトルが頭と尾で結合された三角形の 2 つの辺として配置され、三角形の 3 番目の辺が結果のベクトルの大きさと方向を表すトライアングル ルールを使用して追加できます。
ベクトルの減算は、あるベクトルを別のベクトルに逆算することに似ています。その結果、2つのベクトル間の差は、同じベクトル加算のルールを使用して得られます。
ベクトルにスカラーを掛けると、ベクトルの大きさはスカラーの量によって変化します。
スカラーが正の場合、ベクトルの方向は同じままであり、負の場合、方向は逆になります。

<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。18-19ページ</li>
</ol>

ベクトルの操作

ベクトルは大きさと方向を持つ物理量です。ベクトルの操作には、足し算、引き算、およびスカラー倍があります。

スカラー値によって掛けられたベクトルは、スカラー倍と呼ばれます。得られる結果は異なる大きさを持つ新しいベクトルです。スカラーが正の場合、ベクトルの方向は同じままですが、負の場合はベクトルの方向が反転します。例えば、質量と速度の積は運動量を生み出します。ここでは、スカラー量である質量がベクトル量である速度と掛けられ、ベクトル量である運動量が生まれます。結果の運動量ベクトルの方向は速度ベクトルと同じ方向です。

2つのベクトルはベクトル加法の平行四辺形の法則を使用して幾何学的に足し合わせることができます。この法則を使って足し合わせる場合、2つのベクトルは平行四辺形の隣り合う辺として表されます。両方のベクトルの始点は原点に配置され、それぞれのベクトルに平行な直線が引かれて平行四辺形を形成します。平行四辺形の対角線は原点から平行四辺形の対角の反対の角に引かれ、結果のベクトルを表します。結果のベクトルの大きさと方向は、平行四辺形の対角線の長さと角度を測定することで決定できます。2つのベクトルが同一直線上にある特殊な場合、つまり同じ作用線を持つ場合、平行四辺形の法則を簡略化することができます。この場合、2つのベクトルの代数的またはスカラー的な足し算によって結果ベクトルを得ることができます。

2つのベクトルをグラフィカルに足し合わせるもう1つの方法は、ベクトル加法の三角形の法則です。ベクトル加法の三角形の法則は、2つのベクトルが三角形の2辺として頭から尾まで連結された場合、三角形の第3辺が結果ベクトルの大きさと方向を表すことを述べています。この法則を使ったベクトル加法では、最初のベクトルを原点に配置し、2番目のベクトルを最初のベクトルの尾から頭に連結します。原点から2番目のベクトルの終点までの三角形の第3辺が結果ベクトルを表します。

ベクトルの引き算のプロセスは、1つのベクトルの逆（負）をもう1つのベクトルに加えることを意味します。つまり、ベクトル加法のルールを適用して2つのベクトルの差を得ることができます。

ベクトルは、大きさと方向の両方を持つ物理量です。ベクトル演算には、加算、減算、およびスカラー乗算が含まれます。
ある。。。

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

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

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two-dimensional force system: problem solving

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scalar notation

Vector Components for Force Calculations

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Vector Components in 3D Space

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three-dimensional force system:problem solving

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dot product

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Using the Dot Product to Solve Vector Questions

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cross product

Cross Product Direction and Magnitude

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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 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

Adding and Multiplying Vectors

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