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s say we have two vectors vector u. And vector v. How can we find find two components of vector u.
One of which is parallel to vector v. And other is perpendicular to vector v. So lets draw a picture.
So lets say. This is w1 and this perpendicular to it is w. 2.
How can we find those two components of vector u. Where w1 is parallel to vector v. And w.
To a component of vector u. Is perpendicular or orthogonal to vector v. How can we find these two components.
Now these two components. Theyre not necessarily along the x or y axis. Theyre simply parallel and perpendicular to another vector.
The question is how do we find these two components. Well lets redraw. The picture a different way so once again.
This is going to be a vector u. Now were going to break it up into two components. Lets call this w.
1. And this one w. 2.
And here we have the angle theta. Now. Were gonna say.
This is vector v. Its not as long as w. 1.
But notice that its parallel to it which means that its perpendicular to w. 2. What you need to know is this w.
1. Which im gonna write in red. A component of vector u.
Is the projection of vector u. Onto vector v. Because w.
1. Is the component of vector u. That travels along vector vw2 is the component of vector v.
That is orthogonal to vector u. And the way you calculate it its the vector u. Minus.
The component w1 so this here is the component of vector u. That is parallel to the other vector vector v. And this right here is the component vector u that is perpendicular orthogonal to vector v.
So use a nice formulas.
Thats how we can find that is these two formulas. Thats how we could find a component. That is parallel to another vector and another component vector u.
That is perpendicular to another vector. So now lets work on some practice problems. Now lets work on some practice problems.
Lets start with part a find w1 and the projection of u onto v. Now. Lets say that vector u.
Is 3 comma. 5. And vector v.
Is 2 comma. 4. So with this information how can we find a projection of u onto v.
So heres the formula w1. Which is the projection of u onto v. Its equal to the dot product of u and v divided by the square of the magnitude of vector v times v.
I didnt give that formula to you before but now you have it so using this formula. How can we find w1 so the dot product of u and v. We need some multiply the x components together so thats going to be 3 times.
2. And then we need to multiply the y components. Together.
So thats 5 times. 4. Now the magnitude of vector v.
Is the square. Its the sum of the squares of the x and y components. All within a square root.
But since that is squared the square will cancel the square root and then times. The vector v. Which is 2 comma.
4. Now lets get rid of this and lets perform the operation. So we have 3 times.
2. Which is 6 5 times. 4.
Is 20 that we could cancel the square root. And the square 2. Squared is 4 4.
Squared is 16. And then 6 plus. 20 is 26 4.
Plus. 16. Is 20 and we can reduce that fraction.
If we divide the numerator and the denominator by 2. So 26 over 20 becomes. 13 over 10 now 13 times.
Is 26 4. Times. 13.
Is 52. And then we can also reduce those fractions. 26.
Over 10 reduces to 13 over 5. If you divide both numbers by 2 and the same is true for 52 over 10. If you divide the top and bottom by 2 you get 26 over 5.
So this a vector here is the vector that is parallel to or rather. It is the component of vector u. That is parallel to vector v.
So that is the projection of u onto v. Now the last thing we need to do is find w2. Which is the vector component of u.
Orthogonal or perpendicular to vector v. And w 2. Is going to be vector u minus w.
1. So its three comma five minus 13 over 5 comma 26 over 5. Now lets get common denominators.
So. 3. Is the same as 15 over 5 and.
5. Is the same as 25 over 5. So 15 minus.
13. Is 2 and 25 minus. 26.
Is negative. 1. So this is w.
2. So this is the component of vector u. That is orthogonal or perpendicular to vector v.
Now. Lets work on another. Example.
So. Lets say that vector. U.
Is. 6i. Minus.
3. J. Plus 9.
K. And lets say. That vector.
Is. 4. I minus j.
Plus 8. K. So.
Lets go ahead. And find the two components of vector u. W1.
A w. 2. So lets start with w1.
So just like before w1 is the projection. Vector u onto v. And so its the dot product of u and v divided by the square of the magnitude of v.
Times vector v. So lets start with the dot product of u and v. So lets multiply the x components together.
So thats going to be 6 times. 4. And then lets multiply.
The y components together so thats negative 3 times. A negative. 1 and then multiply the z components so 9 times 8.
And on the bottom. Were gonna have the school of vector v. So the magnitude of v.
Its gonna be the square root of 4 squared plus negative 1 squared. Plus. 8.
Squared and were going to square this times vector v. And that would simplify 6 times. 4.
Is 24 negative. 3 times negative. 1.
Is 3 9. Times. 8.
Is 72. These two will cancel on the bottom. We have.
4 squared. Which is 16 plus. 1.
Squared. Plus. A squared.
Which is 64 24. Plus. 3.
72. Thats 99 16 plus. 64.
Is 80 plus. 1. Thats 81.
Now 99. 81. If we divide both numbers by 9 99.
Divided by 9. Is 11 81. Divided by 9 is 9.
So that gives us this. And so w. 1.
Is going to be 11 9 times. 4. So thats 44 over.
9. I and then minus. 11.
Over. 9. J.
88. 9. K.
So. This is w. 1.
The projection of u onto v. Now. Lets move on to part b so lets calculate w.
2. Which is u minus w. 1.
So we have six i minus 3 j 9 k minus w 1. So lets begin by getting common denominators just like we did before so 6 times. 9 is 54 so lets write 6 as 54 over 9 negative.
3. We can write that as negative 27 over 9 and 9 times 9 is 81. So lets write.
9 as 81 over 9 and now lets distribute the negative sign. So this is gonna be negative 44 over 9 and then positive 11 over 9 and then negative 88 over 9. Now lets combine like terms so lets combine these two first 54 44.
Thats 10. So we have 10 over 9. I next we have negative 27 plus 11.
So that gives us negative 16 and then 81 minus 88. Thats negative. 7.
So this is w 2. The component of vector u. That is orthogonal to vector v.
Thats it .
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