What is the magnitude of the vector
𝐀𝐁 where 𝐴 equals 11, three and 𝐵 equals seven, three?
The magnitude of a vector is its
size or length. So in this case, we need to find
the distance or length between point 𝐴 and point 𝐵. There are several ways of
approaching this problem, we will look at two of them. Our first method will be
graphically, and we will begin by plotting the two coordinates. Point 𝐴 has coordinates 11,
three. Point 𝐵 has coordinates seven,
three. As both points have the same
𝑦-coordinate, the distance from 𝐴 to 𝐵 will be a horizontal distance. To get from 11 to seven, we need to
subtract four. As the magnitude of any vector must
be positive, then the magnitude of 𝐀𝐁 is equal to four.
We could also have calculated the
distance between point 𝐴 and point 𝐵 using one of our coordinate geometry
formulas. The distance between any two points
is equal to the square root of 𝑥 one minus 𝑥 two squared plus 𝑦 one minus 𝑦 two
squared, where our two points have coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦
two. Substituting in our values gives us
𝑑 is equal to the square root of 11 minus seven squared plus three minus three
It doesn’t matter which coordinate
is 𝑥 one, 𝑦 one and which one is 𝑥 two, 𝑦 two. 11 minus seven is equal to four,
and three minus three is zero. As zero squared is equal to zero,
𝑑 is equal to the square root of four squared. As our distance must be positive,
this is equal to four. Once again, we have calculated that
the magnitude of the vector 𝐀𝐁 is four.
Our final question will involve
finding the magnitude of two separate vectors and their sum.