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To put it simply, a vector is essentially an arrow from one point to another.
Vectors can be in three dimensions, or (abstractly) even more!
The base point of a vector is called the tail and the end of the arrow is called the head.
A vector is usually regarded as depending upon its length and direction.
In a contest problem, this idea hides inside a situation you have not seen before. Learning to spot it is the real skill. These are real problems from the AMC 12 and AIME, the national math competitions for middle and high school students, each with a worked solution on LIVE by Po-Shen Loh.
Interactive · 2 questions
In this post, we are going to talk about what vectors are, but before that, lets talk about what the opposite of a vector, a scalar, is.
A scalar is a quantity that can be described using just a magnitude.
For example, time is a scalar, since to describe time we only need a magnitude, like seconds, or minutes, or hours, etc... All of these can be described using a single number - a magnitude.
Now lets define a vector!
A vector is a quantity that is described using a magnitude and a direction.
For example: A displacement of m North. This means that an object moved North 4 meters, the direction is important in this as well, since moving 4 meters South is very different to moving 4 meters North.
Now lets check that we understand!
Which of the following are Vectors?
Okay, so now that we understand the difference between scalars and vectors, lets see how we can represent them!
A vector can be represented using an arrow.
In the image of the vector above, you can see the two things needed clearly. The vector has a magnitude (The length of the vector) and a direction (where the arrow is pointing). We could call the vector something like (with the arrow) or (in bold). If the head is point and the tail is point , then we often call the vector (with the arrow going from the tail to the head ).
Column vectors are another way to describe a vector. On the co-ordinate plane, there is a vector whose tail lies on the origin and the head is at point .
How do you think this can be represented in a column vectors?
Yes! when you write a vector as a column vector, you basically write out the co-ordinates of the head of the vector (if the tail is at the origin, otherwise it wont work).
Read · 1 min
A vector is any quantity that has both a magnitude and a direction. For example, 50 northwest is a vector.
Alternatively, a vector is any quantity that has two or more components in a single, ordered sequence. For example, the quantity <1,2,-5,6,106> is a vector.
The fact that our alternative definition is equivalent to the one above it is easy to show. Simply visualize the quantity as a motion in n-dimensional space, e.g. the quantity <1,2,3> moves the point <-1,-2,-3> in 3-D space to the origin, <0,0,0>. Clearly, this motion has a magnitude and a direction, which means that it is a vector.
