We're given vector a and b
in component form and asked to find
the projection of vector b onto vector a.
So we use the notation projection of vector b
onto vector a.
Before we find this projection
using this formula here below,
let's take a look at the graph
of these two vectors in space.
Again, we're projecting vector b onto vector a.
Vector b is this blue vector.
Vector a is this red vector.
Notice how if we project vector b onto vector a,
because the angle between the two vectors is obtuse,
the projection is in the opposite direction
of vector a, this light green vector here.
So this light green vector here
is the projection of vector b onto vector a.
And if we form this right triangle here,
notice how if we call this projection
vector w sub one and this vector w sub two,
vector w sub one which is a scalar multiple
of vector a plus vector w sub two
is equal to vector b, again, graphed here in blue.
Now let's go ahead and find this vector here,
the projection of vector b onto vector a.
This would be equal to vector b dotted
with vector a divided by the magnitude
of vector a squared times vector a.
So again, vector b
dotted with vector a
divided by the magnitude of vector a squared.
The magnitude of vector a
would be the square root of negative one squared
plus five squared plus negative two squared.
And again, this is squared.
And then times vector a.
So now we'll find the dot product.
We'd have zero times negative one.
That's zero.
Plus negative four times five.
That's negative 20.
Plus negative two times negative two.
So we have plus four.
Divided by, looking at the denominator,
we would have the square root of negative one squared.
That's one.
Plus five squared.
That's 25.
Plus negative two squared.
That's plus four.
So we have the square root of 30,
but that's squared.
Times vector a.
So notice how our scalar multiple
is negative 16 over 30 times vector a.
But this fraction simplifies to negative eight fifteenths.
So we have an x component of eight fifteenths,
a y component of negative 40 fifteenths.
But negative 40 fifteenths simplifies
to the common factor of five
which simplifies to negative eight thirds.
So the y component is negative eight thirds.
And the z component would 16 fifteenths.
This vector is a projection of vector b onto vector a.
Again, going back to our graph just for a moment,
this light green vector here is the projection
of vector b onto vector a which we just found.
I hope you found this helpful.