- What is the slope of ( 2, 5 )
^{T}? m_{1}= 5/2 - What is the slope of ( -5, 2)
^{T}? m_{2}= -2/5 - What is the dot product? (2)(-5) + (5)(2) = 0
- What is m
_{1}times m_{2}? 5/2 * -2/5 = -1

You may recall this from past math classes:

If two lines are perpendicular, then the product of their slopes is -1.

This is another way to look at what goes on when you make a normal vector to a given 2D vector by swapping elements and negating one:

- If
**v**= ( x, y )^{T} - Then
**v'**= ( -y, x )^{T}is orthogonal, - because ( x, y)
^{T}**·**( -y, x ) is -xy + yx = 0. - The slope of ( x, y )
^{T}is y/x. - The slope of ( -y, x )
^{T}is -x/y. - The product of the slopes is y/x * -x/y = -(xy)/(xy) = -1

Are ( -1.5, 6)^{T} and (2, 2)^{T} orthogonal?