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

İki boyutlu bir koordinat sisteminde iki kuvvet vektörünü düşünün.
Bu vektörler, eşzamanlı vektörlerin kuyruklarının bir araya getirildiği ve bir paralelkenar oluşturmak için her vektöre paralel çizgilerin çizildiği paralelkenar yasası kullanılarak eklenebilir.
Elde edilen vektör, kuyruklardan çizgilerin kesişme noktasına bir köşegen çizilerek elde edilebilir.
Benzer şekilde, vektörler, iki vektörün üçgenin iki kenarı baştan kuyruğa birleştirilecek şekilde düzenlendiği ve üçgenin üçüncü tarafının ortaya çıkan vektörün büyüklüğünü ve yönünü temsil ettiği üçgen kuralı kullanılarak eklenebilir.
Vektör çıkarma, bir vektörün tersini diğerine eklemeye benzer. Sonuç olarak, iki vektör arasındaki fark, aynı vektör toplama kuralları kullanılarak elde edilir.
Bir vektör bir skaler ile çarpıldığında, vektörün büyüklüğü skaler miktarın miktarına göre değişir.
Skaler pozitifse, vektörün yönü aynı kalır ve negatifse, yön tersine çevrilir.

<ol>
	<li>Hibbeler, R.C. (2016). Mühendislik Mekaniği \u2012 Statik ve Dinamik. Hoboken, New Jersey: Pearson Prentice Salonu. sayfa 18 -19</li>
</ol>

vektör İşlemleri

Vektörler hem büyüklük hem de yöne sahip fiziksel niceliklerdir . Vektör işlemleri, toplama, çıkarma ve skaler çarpma içerir.

Bir vektör, bir skaler değer tarafından çarpıldığında skaler çarpım adını alır. Elde edilen sonuç, farklı bir büyüklüğe sahip yeni bir vektördür. Eğer skaler pozitifse, vektörün yönü aynı kalır, ancak negatifse, vektörün yönü tersine döner. Örneğin, kütle ve hızın çarpımı momentumu verir. Burada, skaler nicelik kütle, vektör niceliği olan hızla çarpılır ve sonuç vektör niceliği olan momentum elde edilir. Elde edilen momentum vektörünün yönü hız vektörüyle aynı yöndedir.

İki vektör, vektör toplama için paralelkenar yasasıyla geometrik olarak toplanabilir. Bu yasaya göre eklenen iki vektör, bir paralelkenarın bitişik iki kenarı olarak temsil edilir. Her iki vektörün kuyrukları başlangıç noktasına yerleştirilir ve her vektöre paralel çizgiler çizerek bir paralelkenar oluşturulur. Paralelkenarın çaprazı, başlangıç noktasından paralelkenarın karşı köşesine kadar çizilerek sonuç vektörü temsil edilir. Sonuç vektörünün büyüklüğü ve yönü, paralelkenarın çaprazının uzunluğu ve açıları ölçülerek belirlenebilir. Paralelkenar yasası, iki vektörün aynı doğrultuda yani aynı hareket çizgisine sahip olduğu özel bir senaryoda basitleştirilebilir. Bu durumda, iki vektörün cebirsel veya skaler toplamı ile sonuç vektörü elde edilebilir.

İki vektörü grafiksel olarak toplamanın başka bir yöntemi de vektör toplama üçgen yasasıdır. Vektör toplama üçgen kuralı, iki vektörün bir üçgenin iki tarafı gibi baştan kuyruk birleştirilerek düzenlenmişse, üçgenin üçüncü tarafı sonuç vektörünün büyüklüğünü ve yönünü temsil eder. Bu yasayı kullanarak vektör toplama, ilk vektörü başlangıç noktasına yerleştirme ve ikinci vektörü ilk vektöre baştan kuyruk birleştirerek yerleştirme işlemini içerir. Başlangıç noktasından ikinci vektörün sonuna kadar olan üçgenin üçüncü tarafı sonuç vektörünü temsil eder.

Vektör çıkarma işlemi, bir vektörün (negatif) tersini başka bir vektöre ekleyerek yapılır. Bu, iki vektör arasındaki farkı elde etmek için vektör toplama kurallarının uygulanabileceği anlamına gelir.

Vektör İşlemleri

Vektörler, hem büyüklüğü hem de yönü olan fiziksel büyüklüklerdir. Vektör işlemleri toplama, çıkarma ve skaler çarpmayı içerir.
A ...

Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

Force Vectors

Kuvvet Vektörleri

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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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...

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two-dimensional force system: problem solving

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Two-Dimensional Forces: Free-Body Analysis

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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,...

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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

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Resolving Forces in 3D Coordinates

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position vectors

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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 Makine Mühendisliği, denge, kuvvetler, kafes kiriş analizi ve yapısal mekanik gibi temel konuları kapsayan kısa, anlaşılması kolay animasyonlu videolar sunar. Gerçek dünyadaki uygulamalarını keşfedin ve bilim insanlarının önderlik ettiği deneylerle bu kavramların uygulanmasına tanık olun.

mechanical engineering

Adding and Multiplying Vectors

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