00:01
In this question, given the vectors u and v, we are asked to find the projection of u onto v and find the vector orthogonal to the projection so that the sum of the vector z plus the projection vector equals to the vector u.
00:24
And here is a picture.
00:28
Say this is a vector v and somewhere here is the vector u.
00:39
First, we need to find the projection of u onto v, which is going to be this red vector here.
00:52
And then we need to find the vector z, which is the orthogonal vector, and so that the sum of the projection and the vector z gives us the vector u.
01:11
First, let's rewrite the vectors u and v in component form.
01:17
In component form, the vector u equals to negative 6, negative 4, negative 2.
01:27
Recall that the number in front of i goes first, then the second component is the number in front of j, and the last component is the number in front of k.
01:38
For v, there is no component in front, there is no i component, which means it's zero.
01:44
The j component is 4 and the k component is 4.
01:48
Now let's calculate the projection.
01:53
The projection of u onto v, the formula is u .dot product v, divide by the dot product of v by itself, and multiplied by the vector v.
02:09
To calculate the dot product of u and v, we need to multiply the corresponding coordinates of u and v, and then add them up.
02:17
We are going to get negative 6 times 0, the product of the first components, plus negative 4 times 4, which is the product of the 6...