What does it mean for two vectors to be orthogonal?
Their dot product is equal to
It just means they are perpendicular. To find this, take the dot product by taking the first times first plus last times last. If this equals zero, they are orthogonal.
This is also known as the inner product.
For 3D-vectors, do basically the same thing, including the middle term.
Think of two vectors, one pointing straight up, and one pointing straight to the right. Those vectors could be defined like so:
Since they form a right angle, they are orthogonal. Taking the dot product we find...
Essentially, they are at right angles to one another and their dot product is zero.
If they are also of length
A set of
If you form an
Such a matrix represents an orthogonal transformation - preserving angles and distances - essentially a combination of rotation and possible reflection.