In this article, we are going to testimonial the vector. Vectors--unlike basic numbers (scalars) that have actually only a magnitude--have both a magnitude (length) and also direction. Us will check out how to stand for vector quantities, and how to add and subtract them.

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Key Terms


Individual numbers--that is, worths that have only (positive or negative) magnitude--are called scalars. The numbers 0, –3, π, i, 1.3, e, and so on space all instances of scalars. Another type of value that is often valuable in mathematics is the vector. A vector is a quantity that has both size and direction. In this article, we will consider some the the mathematical features of vectors. Vectors have considerable applications in, because that example, physics.

Introduction come Vectors

To recognize the difference between a scalar and also a vector, it helps to think of physical examples. Take into consideration temperature, for instance. You have the right to use a thermometer to measure up the wait temperature at various locations. In each case, you get some number (and a unit)--say, 65°F. This is a magnitude, however it has no direction connected with it; that is hence a scalar quantity. Now, consider measurements the the wind at these same locations. Once you measure the wind, you would most likely measure both the speed and also the direction. Thus, your wind dimensions constitute a vector. We might express this vector as an arrowhead pointed in the direction of the wind, v the length of the arrow being proportional come the wind speed. Below is an illustration of 2 wind dimensions taken at various points; the arrows represent the vectors associated with this measurements.


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Vectors have actually a magnitude and also direction, however they perform not have an assigned ar per se. The is, as long as the direction and length of the "arrow" is maintained, we can move it almost everywhere we desire without changing it. This is an essential characteristic that will let us work generally with vectors.

A representation of Vectors

Our first task is to discover a method to cleanly and consistently represent vectors. Graphically, this is simple: due to the fact that we deserve to move a vector everywhere we want, let"s always position the "tail" the the vector in ~ the origin of the coordinate plane. (Note that the "head" and also "tail" of a vector are defined as presented below.)

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Now, with the tail the the vector put at the origin (remember, we have the right to move the vector anywhere as long as we preserve its direction and length), we deserve to quantify it together the coordinates of the head. An instance is shown listed below for vector v. (Note that to identify symbols representing vectors native those representing scalars, we use boldface. One more common technique is to usage a little arrow over the symbol: for instance, vector

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.)


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Thus, the vector v is simply the coordinates of the suggest at (2, 3). Keep in mind that every one of the vectors shown listed below are equal to (2, 3)--our convention is that the vector is defined by the works with of the suggest at the head only as soon as its tail is positioned at the origin.


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Although us have presented the vector just in two dimensions, this strategy can be generalised to any number of dimensions. For instance, in three dimensions, a vector will have actually the form (x, y, z). Every one of the properties of two-dimensional vectors have the right to be easily extended to 3 dimensions.

But just how do we go about "moving" the vector, from a number perspective? for instance, say a vector v has actually its head in ~ (3, 2) and its tail at (1, 4).


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The price lies in translating (or moving) both the head and also the tail over an indistinguishable distance and also in the same direction. This translate into should an outcome in the tail that the vector relocating to the origin--a simple process that requires subtracting each tail name: coordinates from itself. In the above example, the an outcome is (3 – 3, 2 – 2) = (0, 0). To translate the head, similarly subtract the tail coordinates from the head coordinates--this satisfies our criterion the the translation have actually a fixed distance and direction. Thus, the head need to be moved as follows: (1 – 3, 4 – 2) = (–2, 2). Thus, in general, to uncover the worth of an arbitrarily put vector, subtract the collaborates of the tail from the works with of the head. This procedure is portrayed below.

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Note the the vector (0, 0), sometimes dubbed the zero vector, has actually a length of 0 but no identified direction. (That is, no issue what direction friend pick, the zero vector is the same.)

Practice Problem: determine the value of every vector displayed in the graph below.


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Solution: In each case, you can find the coordinate expression for the vector by individually the tail works with from the matching head coordinates. This works also when the tail is at the origin, i m sorry has works with (0, 0). But if the tail is at the origin, the vector is additionally simply equal to the collaborates of the head. If it help you, redraw the vectors through tails positioned in ~ the origin.

a = (–1, 4)

b = (–3, –3)

c = (3 – 3, 2 – 0) = (0, 2)

d = (3 – 2, –4 – <–1>) = (1, –3)

Adding and also Subtracting Vectors


As with scalars, we can add and subtract vectors. The process is similar, however with one or 2 caveats. To add or subtract 2 vectors a and b, include or subtract corresponding works with of the vector. The is, whereby a and also b are defined as follows, below are the rule for addition and subtraction.

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Note that just like scalars, enhancement of vectors is commutative, yet subtraction is not. Graphically, we add two vectors a and also b by placing the tail that b at the head the a and also then creating a brand-new vector beginning from the tail of a and ending at the head that b. The works with of this new vector are identified in the same way as before: by placing its tail in ~ the origin. This procedure is illustrated listed below for vectors a = (4, 1) and b = (-1, 2).

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Note that

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Subtracting vectors complies with basically the very same procedure together addition, except the vector being subtracted is "reversed" in direction. Think about the very same vectors a and also b together above, other than we"ll calculation ab. (Note that this is the very same as

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, whereby –b has the same length as b yet is the contrary in direction.)