High School What are the values in a vector?

[The Subject]

I'm trying to understand the concept of vectors. Vectors have magnitude and a direction. When I read vector with some values
<br /> \textbf{x} = \left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right) = \left(\begin{array}{c}1\\2\\3\end{array}\right)<br />
I'm not sure what these values are. Are the values points in the vector? Would I be wrong to imagine in a x,y graph, the values in this matrix are the values of y?

 

[jedishrfu]

This diagram may help:

So the vector a has components a1, a2, and a3

This is the most common usage. However, it doesn't have to be an orthogonal set of axes. It depends on the problem and the notational conventions you're using for describing your vector.

By using a numbered set of axes you can generalize things to n-dimensional space.

 

[drvrm]

I'm not sure what these values are. Are the values points in the vector?

as your vectors are written as a column matrix with three elements it should be a three dimensional vector-then you can interpret the numbers!

 

[The Subject]

The diagram is super helpful!
This makes sense since x1 + x2 + x3 is 3 dimensional.

 

[LCKurtz]

The diagram is super helpful!
This makes sense since x1 + x2 + x3 is 3 dimensional.

The picture represents a 3 dimensional object. The arrow representing the vector is 3 dimensional. $x_1+x_2+x_3$ is just a number.

 

[HallsofIvy]

A vector written as you have it make no sense without first having said what those number mean! A vector is defined by its 'direction and length'. A vector having a fixed direction and length 1 can be written as (1, 0, 0) or (0, 1, 0) or (0, 0, 1) or \left(\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\right), etc. depending upon what basis or coordinate system it is given in.

 

[Ssnow]

Vectors are element of a vector space $V$. The axioms of a vector space are given after fixed the set of scalars (or generally a field) $\mathbb{K}$. This can be $\mathbb{R},\mathbb{C}, ...$. The components of your vectors are elements of $\mathbb{K}$ ...

 

[Inventive]

First take the 2 dimension case. Imagine the a baseball diamond where x1 = x , X2 = y.
For any given pair of points (x,y) in the xy plane, a line segment can be drawn in the x direction of length x
and another line segment can be drawn of length y in the y direction. With the tails of each line segment
touching, a new line segment can be drawn between the x and y line segments starting at the adjoining tails
Who's length is the sum of lengths of each x and y line segment. The angle (direction in the plane) of thIs
vector can be found using x and y and trigonometry. Since this vector is bounded sorry far by by x,y only, now
introduce x3 = z. Raise this vector the length of z units upward or downwards. This now defines the vector in 3 space
that is bounded by the 3 planes xy, xz, and yz. The angle of the vector relative to xz or yz is the re-application
of trigonometry using (x,z) or (y,z).