This is x and this is y,

we are interested in the angle omega between those.

If we use a dot product as the inner product,

we get that the cosine of omega is

x transpose y divided by the square root of x transpose x times y transpose y,

which is three divided by the square root of 10.

This means the angle is approximately 0.32

radians or 18 degrees.

Intuitively, the angle between two vectors tells us how similar their orientations are.

Let's look at another example in 2D again with a dot product as the inner product.

We're going to look at the same vector x that we used to have before,

so x equals one,

one which is this vector over here,

and now choose y to be minus one and plus one and this is this vector.

Now we're going to compute the angle between these two vectors and we

see that the cosine of this angle between x and y

is with a dot product x transpose times y divided by the norm of x times the norm of y.

And this evaluates to zero.

This means that omega is pi over two in radians,

if you want to say this in degrees,

we have 90 degrees.

This is an example where two vectors are orthogonal.

Generally, the inner product allows us to characterise orthogonality.

Two vectors, x and y,

where x and y are non-zero vectors,

are orthogonal if and only if their inner product is zero.

This also means that orthogonality is defined with respect to inner product.

And vectors that are orthogonal with respect to

one inner product do not have to be orthogonal with respect to another inner product.

Let's take these two vectors that we just had where

the dot product between them gave that they are orthogonal,

but now we are going to choose a different inner product.

In particular, we are going to choose

the inner product between x and y to be x transpose times the matrix two,

zero, zero, one, times y.

And if we choose this inner product,

it follows that the inner products between x and y is minus one.

This means that the two vectors are not

orthogonal with respect to this particular inner product.

From a geometric point of view,

we can think of two orthogonal vectors as two vectors that are most

dissimilar and have nothing in common besides the origin.

We can also find a basis of

a vector space such that the basis vectors are orthogonal to each other.

That means, we get the inner product between b_i and b_j is

zero if i is not the same index as j.

And we can also use the inner product to normalise these basis vectors.

That means, we can make sure that every b_i has length one.

Then we call this an orthonormal basis.

In this video, we discussed how to compute angles between vectors using inner products.

We also introduced the concept of orthogonality,

and so that vectors maybe orthogonal with respect to one inner product,

but not necessarily if we change the inner product.

We will be exploiting orthogonality later on in the course.

If we have a vector and we want to compute

the smallest difference vector to any point on a line that does not contain the vector,

then we will end up finding a point on the line such as the segment

between the point and the original the vector is orthogonal to that line.