Section outline

  • A person shooting a basketball towards a hoop in a gymnasium.

    Successfully shooting a basketball requires a subconscious understanding of the involved in how the basketball moves through the air. The vertical and horizontal vectors must be perfectly organized if the ball is to pass through the basket.
    ::成功拍摄篮球需要潜意识了解篮球如何在空气中移动。 如果球要穿过篮子,则垂直和横向矢量必须完全组织起来。

    Graphical Methods Vector Addition
    ::图形方法矢量添加

    In physics , a quantity, such as mass, length, or that is completely specified by its magnitude and has no direction is called a scalar .  A vector , on the other hand, is a quantity possessing both magnitude and direction. A vector quantity is typically represented by an arrow-tipped line segment. The length of the line, drawn to scale, represents the magnitude of the quantity. The direction of the arrow indicates the direction of the vector. Not only can vectors be represented graphically, but they can also be added graphically.
    ::在物理学中,一个数量,如质量、长度,或完全由质量、长度指定,且没有方向的,被称为标量。另一方面,矢量是一个既具有音量又具有方向的数量。矢量通常由箭头倾斜的线段表示。线条的长度按缩放来表示数量大小。箭头的方向表示矢量的方向。不仅可以用图形表示矢量,还可以用图形添加。

    For one dimensional vector addition , the first vector is placed on a number line with the tail of the vector on the origin.  The second vector is placed with its tail exactly on the arrow head of the first vector.  The sum of the two vectors is the vector that begins at the origin and ends at the arrow head of the final added vector.
    ::对于一个维矢量添加,第一个矢量将放置在与来源方矢量尾部的数条线上。第二个矢量的尾巴完全放在第一个矢量的箭头上。两个矢量的总和是从来源开始的矢量,最后添加的矢量的箭头的矢量。

    Consider the following two vectors.
    ::考虑以下两个矢量。

    Red and blue arrows representing two vectors with magnitudes 11 and -3 on a number line.

    The red vector has a magnitude of 11 in the positive direction on the number line. The blue vector has a magnitude of -3, indicating 3 units in the negative direction on the number line. In order to add these two vectors, we place one of the vectors on a number line and then the second vector is placed on the same number line such that its origin is on the arrow head of the first vector.
    ::红色矢量在数字线上正方向为11。 蓝色矢量在 - 3 中, 表示数字线上负方向为3 个单位。 为了添加这两个矢量, 我们将其中一个矢量放在数字线上, 然后将第二个矢量放在同一个数字线上, 以便其来源在第一个矢量的箭头上 。

    Number line showing vector addition: red vector (11) and blue vector (-3) resulting in purple vector.

    The sum of these two vectors is the vector that begins at the origin of the first vector (the red one) and ends at the arrow head of the second (blue) vector. So the sum of these two vectors is the purple vector, as shown below.
    ::这两个矢量的总和是始于第一个矢量来源的矢量(红色矢量),以第二个(蓝色)矢量的箭头为终点。这两个矢量的总和是紫矢量,如下文所示。

    Vector Addition: Red And Blue Vectors Combined To Form A Purple Vector.

    The vector sum of the first two vectors is a vector that begins at the origin and has a magnitude of 8 units in the positive direction.  If we were adding three or four vectors all in one dimension, we would continue to place them head to toe in sequence on the number line.  The sum would be the vector that begins at the beginning of the first vector and goes to the ending of the final vector.
    ::前两个矢量的矢量总和是一个从来源开始的矢量,在正方向有8个单位的大小。如果我们在一个维度中添加3个或4个矢量,我们将继续在数字线上按顺序排列它们头部的脚趾。这个总和将是第一个矢量开始的矢量,然后到最后矢量的终点。

    Adding Vectors in Two Dimensions
    ::在两个维中添加矢量

    In the following image, vectors  A and  B represent the two displacements of a person who walked 90. m east and then 50. m north. We want to add these two vectors to get the vector sum of the two movements.
    ::在下图中,矢量A和矢量B代表一个人在东行90米,北行50米的两次迁移。我们要加上这两个矢量,以获得两种移动的矢量总和。

    Vectors representing displacements of 90 m east and 50 m north for vector addition.

    The graphical process for adding vectors in two dimensions is to place the tail of the second vector on the arrow head of the first vector as shown above.
    ::在两个维度中添加矢量的图形进程是将第二个矢量的尾部置于上面所示第一个矢量的箭头上。

    The sum of the two vectors is the vector that begins at the origin of the first vector and goes to the ending of the second vector, as shown below.
    ::两个矢量的总和是始于第一个矢量来源的矢量,然后到第二个矢量终点的矢量,如下文所示。

    Vector addition illustration showing angles and directions of two vectors in 2D.

    If we are using totally graphic means of adding these vectors, the magnitude of the sum would be determined by measuring the length of the sum vector and comparing it to the original standard. We would also use a compass to measure the angle of the summation vector.
    ::如果我们使用完全图形手段来添加这些矢量,那么总和的大小将通过测量矢量的总长度并将其与原始标准进行比较来确定。我们还将使用一个指南针来测量总和矢量的角度。

    If we are using calculation means, we can divide 50. m by 90. m and determine inverse tangent of the dividend. The result of 29.05 indicates the angle of 29° north of east. The length of the sum vector can also be determined mathematically by the Pythagorean theorem,  a 2 + b 2 = c 2 .  In this case, the length of the hypotenuse would be the square root of (8100 + 2500) or 103 m.
    ::如果我们使用计算方法,我们可以将50米除以90米,并确定股息的反正切值。29.05的结果显示东面以北29度为角。总矢量的长度也可以由Pytagorean定理(a2+b2=c2)数学确定。在这种情况下,下限的长度是平方根(8100+2500)或103米。

    If three or four vectors are to be added by graphical means, we would continue to place each new vector head to toe with the vectors to be added until all the vectors were in the coordinate system. The sum vector is the vector from the origin of the first vector to the arrowhead of the last vector. The magnitude and direction of the sum vector can be measured.
    ::如果要用图形手段添加三或四个矢量,我们将继续将每个新的矢量头指向每个矢量指向添加的矢量,直到所有矢量都在坐标系统中。总和矢量是从第一个矢量的源头到最后一个矢量的箭头的矢量。可以测量总矢量的大小和方向。

    A person using a phone for navigation while sitting in a car.

    Have you ever used a phone app that provides directions or a navigation system in your car? These programs help you get from Point A to Point B by breaking it down into a series of left and right turns that exemplify many of the graphical methods of described above. The navigation systems in self-driving cars are even more advanced. Continue to practice vector addition by helping  a driverless car get to its destination in the following simulation:
    ::您是否在车上使用过提供方向或导航系统的电话应用程序? 这些程序帮助您从A点到B点, 将其分成一系列的左转和右转, 说明许多上述图形方法。 自驾驶汽车的导航系统甚至更先进。 继续练习矢量添加, 帮助无司机的汽车在以下模拟中到达目的地 :

    Summary
    ::摘要

    • Scalars are quantities, such as mass, length, or speed, that are completely specified by magnitude and have no direction.
      ::飞毛腿是质量、长度或速度等数量,按数量完全指定,没有方向。
    • Vectors are quantities possessing both magnitude and direction and can be represented by an arrow; the direction of the arrow indicates the direction of the quantity and the length of the arrow is proportional to the magnitude.
      ::矢量指具有大小和方向并可由箭头表示的矢量;箭头的方向指数量的方向,箭头的长度与数量成正比。
    • Vectors that are in one dimension can be added arithmetically.
      ::一个维度中的矢量可以用算法添加。
    • Vectors that are in two dimensions are added geometrically.
      ::分两个维的矢量是几何相加的。
    • When vectors are added graphically, graphs must be done to scale and answers are only as accurate as the graphing.
      ::当矢量用图形方式添加时,必须用比例尺绘制图表,答案只能与图形一样准确。

    Review
    ::回顾

    1. On the following number line, add the vector 7.5 m/s and the vector -2.0 m/s.
    ::1. 在以下数字行中,加上矢量7.5m/s和矢量-2.0m/s。

    Number line showing vectors: 7.5 m/s and -2.0 m/s for vector addition.

    2. On a sheet of graph paper, add a vector that is 4.0 km due east and a vector that is 3.0 km due north.
    ::2. 在图表纸页上,加上东经4.0公里的矢量和北经3.0公里的矢量。

    Explore More
    ::探索更多

    Use this resource to answer the  questions that follow.
    ::使用此资源回答下面的问题 。

    1. What is a resultant?
      ::其结果是什么?
    2. What are the steps necessary to add vectors in two dimensions?
      ::在两个层面增加矢量的必要步骤是什么?