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Vector - Simple English Wikipedia, the free encyclopedia

  • ️Wed Mar 25 2020

From Simple English Wikipedia, the free encyclopedia

This article is about a mathematical concept. For other uses, see Vector (biology).

A vector

A vector is a mathematical object that has a size, called the magnitude, and a direction. It is often represented by boldface letters (such as {\displaystyle \mathbf {u} }, {\displaystyle \mathbf {v} }, {\displaystyle \mathbf {w} }), or as a line segment from one point to another (as in {\displaystyle {\overrightarrow {AB}}}).[1][2]

For example, a vector would be used to show the distance and direction something moved in. When asking for directions, if one says "Walk one kilometer towards the North", that would be a vector, but if they say "Walk one kilometer", without showing a direction, then that would be a scalar.

We usually draw vectors as arrows. The length of the arrow is proportional to the vector's magnitude. The direction in which the arrow points to is the vector's direction.[3]

  • John walks north 20 meters. The direction "north" together with the distance "20 meters" is a vector.
  • An apple falls down at 10 meters per second. The direction "down" combined with the speed "10 meters per second" is a vector. This kind of vector is also called velocity.
  • The distance between two places is 10 kilometers. This distance is not a vector because it does not contain a direction.
  • The number of fruit in a box is not a vector.
  • A person pointing is not a vector because there is only a direction. There is no magnitude (the distance from the person's finger to a building, for example).
  • The length of an object.
  • A car drives at 100 kilometers per hour. This does not describe a vector, as there is only a magnitude, but no direction.
  • Displacement is a vector. Displacement is the distance that something moves in a certain direction. A measure of distance alone is a scalar.
  • Force that includes direction is a vector.[3]
  • Velocity is a vector, because it is a speed in a certain direction.[3][4]
  • Acceleration is the rate of change of velocity. An object is accelerating if it is changing speed or changing direction.
Head-to-tail Addition

The Head to Tail method of adding vectors is useful for doing an estimate on paper of the result of adding two vectors. To do it:

  • Each vector is drawn as an arrow with an amount of length behind it, where each unit of length on the paper represents a certain magnitude of the vector.
  • Draw the next vector, with the tail(end) of the second vector at the head(front) of the first vector.
  • Repeat for all further vectors: Draw the tail of the next vector at head of the previous one.
  • Draw a line from the tail of the first vector to the head of the last vector - that's the resultant(sum) of all the vectors.

It's called the "Head to Tail" method, because each head from the previous vector leads in to the tail of the next one.

[clarification needed]

Using the component form to add two vectors literally means adding the components of the vectors to create a new vector.[5] For example, let a and b be two two-dimensional vectors. These vectors can be written in terms of their components.

{\displaystyle \mathbf {a} =(a_{x},a_{y})}

{\displaystyle \mathbf {b} =(b_{x},b_{y})}

Suppose c is the sum of these two vectors, so that c = a + b. This means that {\displaystyle \mathbf {c} =(a_{x}+b_{x},a_{y}+b_{y})}.

Here is an example of addition of two vectors, using their component forms:

{\displaystyle \mathbf {a} =(3,-1)}

{\displaystyle \mathbf {b} =(2,2)}

{\displaystyle {\begin{aligned}\mathbf {c} &=\mathbf {a} +\mathbf {b} \\&=(a_{x}+b_{x},a_{y}+b_{y})\\&=(3+2,-1+2)\\&=(5,1)\end{aligned}}}

This method works for all vectors, not just two dimensional ones.

The dot product is one method to multiply vectors. It produces a scalar. It uses component form:

{\displaystyle {\begin{aligned}\mathbf {a} &=(2,3)\\\mathbf {b} &=(1,4)\\\mathbf {a} \cdot \mathbf {b} &=(2,3)\cdot (1,4)\\&=(2\cdot 1)+(3\cdot 4)\\&=2+12\\&=14\end{aligned}}}

The cross product is another method to multiply vectors. Unlike dot product, it produces a vector. Using component form:

{\displaystyle \mathbf {a} \times \mathbf {b} =|\mathbf {a} ||\mathbf {b} |\sin(\theta )\mathbf {n} }

Here, {\displaystyle |\mathbf {a} |} means the length of {\displaystyle \mathbf {a} }, and {\displaystyle \mathbf {n} } is the unit vector at right angles to both {\displaystyle \mathbf {a} } and {\displaystyle \mathbf {b} }.

To multiply a vector by a scalar (a normal number), you multiply the number by each component of the vector:

{\displaystyle c\,\mathbf {x} =(c\,x_{1},c\,x_{2},...,c\,x_{n})}

An example of this is

{\displaystyle {\begin{aligned}c&=5\\\mathbf {x} &=(3,4)\\c\,\mathbf {x} &=(5\cdot 3,5\cdot 4)\\&=(15,20)\end{aligned}}}

  1. ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-19.
  2. ↑ Weisstein, Eric W. "Vector". mathworld.wolfram.com. Retrieved 2020-08-19.
  3. ↑ 3.0 3.1 3.2 "Vectors". www.mathsisfun.com. Retrieved 2020-08-19.
  4. ↑ "vector | Definition & Facts". Encyclopedia Britannica. Retrieved 2020-08-19.
  5. ↑ "1.1: Vectors". Mathematics LibreTexts. 2013-11-07. Retrieved 2020-08-19.