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

考虑二维坐标系中的两个力矢量。
这些矢量可以通过平行四边形法则进行相加，即将共点矢量的起点连接在一起，并作平行于各矢量的直线，构成一个平行四边形。
通过从尾部向直线交点画一条对角线，即可得到合成矢量。
类似地，可以使用三角形法则来合成矢量，即将两个矢量首尾相接构成三角形的两条边，三角形的第三条边即表示合矢量的大小和方向。
向量减法类似于将一个向量的反向向量加到另一个向量上。因此，两个向量之间的差值可通过相同的向量加法规则得到。
当向量与标量相乘时，向量的大小会按标量的数值发生变化。
如果标量为正，则向量的方向保持不变；如果标量为负，则方向反转。

<ol>
	<li>希伯勒，RC（2016 年）。工程力学 \u2012 静力学和动力学。新泽西州霍博肯:Pearson Prentice Hall。页码: 18 -19</li>
</ol>

向量运算

向量是具有大小和方向的物理量。向量的运算包括加法、减法和标量乘法。 向量乘以标量的值被称为标量乘法。得到的结果是一个大小不同的新向量。如果标量为正数，则向量的方向会保持不变，如果为负数，那么向量的方向就会反转。例如，质量和速度的乘积能够产生动量。在这里，质量与速度矢量进行相乘，从而产生了动量矢量。合成动量矢量的方向与速度矢量的方向是相同的。 两个向量可以通过使用向量加法的平行四边形法则进行几何相加。使用这个法则相加时，其中的两个向量可以表示为平行四边形的相邻边。两个向量的起点共同被放置在原点处，然后画出与每个向量平行的线段所形成的平行四边形。然后从原点到平行四边形的对角线上画出其结果向量。可以通过测量平行四边形对角线的长度和角度来确定结果向量的大小和方向。在特殊情况下，平行四边形法则可以简化为两个向量共线的情况，这意味着它们具有相同的作用线。在这种情况下，可以通过代数或数量的加法来得出其结果向量。 另一种图形加法方法是三角形相加法则。向量的三角形相加法则规定，将两个向量头尾相接，可以构成一个三角形的两条边，那么三角形的第三边则表示了结果向量的大小和方向。使用这个法则对向量进行相加时，需要将第一个向量放在原点的位置，并将第二个向量从第一个向量的尾部进行连接。从原点到第二个向量末端所组成三角形的第三边则表示结果向量。 向量减法是将一个向量取反（负向量），然后与另一个向量相加，这样便可以应用向量加法法则来得出两个向量之差。

向量是具有大小和方向的物理量。向量运算包括加法、减法和标量乘法。  
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Education

教育

JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Force Vectors

力矢量

KEY TERMS AND CONCEPTS

关键术语与概念

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

scalar and 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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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...

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Resolving Forces in a Plane

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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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cartesian vector notation

Vector Components in 3D Space

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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

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three-dimensional force system:problem solving

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Resolving Forces in 3D Coordinates

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position vectors

Using Position Vectors to Locate Points

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dot product

Using Dot Product to Find Work and Angle

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dot product: problem solving

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 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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