Given the points $A=(1,0,2)$ and $B=(3,-1,0)$ what does $\overrightarrow{AB}$ mean?

Question

Given the points $A=(1,0,2)$ and $B=(3,-1,0)$ what does $\overrightarrow{AB}=(2,-1,-2)$ mean?

Answer 1

$$\vec{AB}$$ means the vector $$AB$$ it is $$(3-1,-1-0,0-2)=(2,-1,-2)$$ if $A$ is given by $$A[x_A,y_A,z_A]$$ and $$B[x_B,y_b,z_B]$$ then is $$\vec{AB}=(x_B-x_A,y_B-y_A,z_B-z_A)$$

Answer 2

$\vec{AB}$ denotes an arrow which goes from the point $A=(a_1,a_2,a_3)$ to the point $B=(b_1,b_2,b_3)$.

This arrow has the same direction and the same length with the vector $(b_1-a_1,b_2-a_2,b_3-a_3)$.

Answer 3

Something like $A=(1,0,2)$ are the coordinates of a point called $A$, just like $B=(3,-1,0)$ are the coordinates of point $B$.

Let's call the coordinates of a point in general $(x,y,z)$, so $x$ is the position of the point in the $x$ direction. Let's also say that there's a positive and negative direction for $x$, and we'll say that $+x$ is to the east and $-x$ is to the west.

Similarly, $+y$ is to the north, and $-y$ is to the south. Also similarly, $+z$ is up, and $-z$ is down.

When we go from point $A$ to point $B$, the $x$ coordinate increases by $2$. This is the $x$ value in $\vec{AB}$. The $y$ coordinate decreases by $1$ (goes from $0$ to $-1$), so the $y$ value in $\vec{AB}$ is $-1$. The $z$ coordinate decreases by $2$, so its value in $\vec{AB}$ is $-2$.

A general formula for the vector from $A$ to $B$, then is $(x_B-x_A, y_B-y_A, z_B-z_A)$.

Graphically, this is an arrow starting on point $A$, and landing on point $B$.

Answer 4

Suppose that you have two points, say $A=(a_1, a_2, a_3)$ and $B = (b_1, b_2, b_3)$. Then $\overrightarrow{AB}$ represents an arrow that originates at $A$ and terminates at $B$. This arrow is a representative of a class of vectors that act on three dimensional space. Vectors act by moving points around in some fixed manner. In this case, the arrow takes the first coordinates from $a_1$ to $b_1$, hence it changes the first coordinate by adding $b_1 - a_1$. Similarly, the arrow adds $b_2 - a_2$ to the second coordinate, and $b_3 - a_3$ to the third coordinate.

This gives us a general the general formula $$ \overrightarrow{AB} = \langle b_1 - a_1, b_2 - a_2, b_3 - a_3 \rangle. $$

In the case that $A = (1,0,2)$ and $B = (3,-1,0)$, we can use this formula to get $$ \overrightarrow{AB} = \langle 3-1, -1-0, 0-2 \rangle = \langle 2, -1, -2\rangle. $$ Again, this object $\overrightarrow{AB}$ is actually a representative of a class of arrows, called a vector. This vector acts on points in three dimensional space by increasing the first coordinate by 2, decreasing the second coordinate by 1, and decreasing the third coordinate by 2. In particular, acting on $A$: $$ \overrightarrow{AB} + A = \langle 2, -1, -2 \rangle + (1,0,2) = (2+1, -1+0, -2+2) = (3, -1, 0) = B. $$ Note that the "$+$" symbol is being abused a little here, since we aren't really adding two things together (vectors and points aren't the same kind of object, so it doesn't exactly make sense to add them), but given the action of a vector on a point, it seems a reasonable abuse.