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Hello, hope you're doing well.
00:03
So for this problem, we're asked to prove that these two expressions are equivalent.
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So before we dive into the problem, let's go and review how to find the cross product and dot product of two vectors.
00:14
So let's start with cross products.
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Let's say we have the cross product of v cross w.
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The way we're going to find that is going to take the determinant of a 3 by 3 matrix, in which the first row is your unit vector's i, j, and k.
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The second row consists of the components of your first vector, which in this case is v1, v2, and v3.
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The third row will consist of the components of your second vector, which are w1, w2, and w3.
00:42
Okay.
00:44
So, and then the way you simplify this and take the determinant of this 3x3 matrix is you take your unit vector i, multiply it by the determinant of this 2x2 matrix, and subtract your unit vector j multiplied by the determinant of this 2x2 matrix, and then add the unit vector k multiplied by the determinant of this 2x2 matrix.
01:06
Then once you get that, you simplify it and you end up with the answer for v -cross -w.
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So, and then if you want to take the dot product of two vectors, let's say u .d .w, what you do is you find the, you take the x components of both of those vectors and multiply them together.
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And you take the product of the y components of those vectors and the product of the z components of those factors.
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Those gives you, that gives you three separate values that you then add together to get your dot product.
01:36
All right, so now keeping that in mind, let's go ahead and prove this.
01:39
So we're going to start out by solving for the left -hand side of this equation.
01:44
So let's go and take a solve for v cross -w first.
01:49
So we already have our 3x3 matrix determinant of our 3x3 matrix set up here.
01:55
So now we're just going to go and take the determinant.
01:57
So we're going to take our i unit vector, multiplied by the determinant of this 2x2 matrix here, which is v2, v3, w2, w2, w3.
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So these vertical lines here, meaning we're taking the determinant of the matrix inside.
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And we're subtracting our unit vector j multiplied by the determinant of this 2x2 matrix here, which is v1, v3, w1, w3.
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We're going to add the k unit vector multiplied by the determinant of this.
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2x 2x which is v1, v2, w1, w2.
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So now before we simplify this, we need to review how to find the determinant of the 2x2 matrix.
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So if you have a 2 by 2 matrix with components a, b, c, and d, then the determinant is equal to a times d minus b times c.
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So keeping that in mind, we can go and simplify this.
02:58
So we've got i multiplied by a times d, is v2 w3, minus v times c, that's minus v2, or minus v3 w2.
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We've got minus j multiplied by ad is v1w3, minus bc, that's v3w1.
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We're going to add our k vector multiplied by ad, that's v1w3.
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Minus bc, it's v2w1.
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Okay, so now we've got our cross -products.
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Let's go in write out our cross -product in bracketed form.
03:51
So you have v2w3 minus v3w2 minus v1w3 minus v3w1.
04:02
They have v1 w2 minus v2 w1 w1 so now let's find u cross v cross w so remember our u vector is equal to or it's just going to be u1 u2 u3 okay so setting this up we have set up our 3 by 3 matrix so we have i j k the second row we have our components of our u vector, that's u1, u2, u3.
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And third row we have our components of our v cross -w vector, which is v2w3 minus v3w2.
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We're a little hard to fit, but let's go in multiply this by negative 1, so we end up with v3w1 minus v1w3.
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That our k vector is v1 w2 minus v2 w1.
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It's a little crowded in there, but this is the third row.
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So we have these are our three components.
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Okay? so now we have, so now we need to take the determinant.
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So we're going to, of this determinant of this 3x3 matrix.
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So we've got i multiplied by the determinants of the determinants of the determinants of the this matrix right here.
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So i multiplied by the determinant of u2.
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Bottom here is v3w1 minus v1w3.
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And it's u3, v1w2, v2, v2w1.
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Okay, minus j multiplied by the determinant of this matrix there.
06:06
So it's u1 here, then v2w2, minus v3 w2, then it's u3 v1 w2, then it's u3 v1 w2 minus v2 w1, taking the determinant of this, and then plus, finally, plus k, multiply by the determinant of, it's going to be the determinant of this matrix.
06:36
So it's going to be u1 on the top, and here we've got v2 w3 minus v3 w2, and then next is going to be u2 here, then v3w1 minus v1w3.
06:55
Okay, so remember the determinant of a matrix is going to be ad minus bc.
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So we're going to have i multiplied by a, ad here.
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It's going to be u2, v1w2 minus v2w1.
07:17
Let me just check something real quick to see how we, okay, no, never mind.
07:22
It's still working.
07:22
Okay.
07:23
So minus b -c, it's minus u3, v3w1, minus v1w3.
07:35
We have minus j.
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A, d is this quantity here.
07:43
So it's u1 times v1w2 minus v2 w -1.
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Then minus b2 .c2.
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That's u3 times this quantity.
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So minus u3 times v2 w3 minus v3 w2.
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Okay, almost there.
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Now plus k multiplied by a d.
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It's right here, so u1 times v3 w1 minus v1 w3 minus bc that's u2 times v2 w3 minus v3w2.
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Okay.
08:29
So let's just write this all out more concisely.
08:32
So for i, we have u2 times v1w2 minus v2 w1.
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Oops.
08:42
Minus u3 v3w1 minus v1w3.
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I.
08:51
Then j, what's playing it by negative 1, or components by negative 1, plus u3 v2w3 minus v3w2 minus v3w2 minus u1 minus u1 w2 minus v2 w1j and plus our k here, which is u1 v3w1 minus v1w3 minus v1w3 minus v2 v2 w3 minus v2 w3 and 2 v2 okay.
09:30
Okay.
09:31
So this right here is u cross v cross w.
09:36
Okay, so let's check to make sure that this quantity here, we get the same thing.
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So first we just get u .w.
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It's our first task.
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U.
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Dot w is equal to u1, u2, u3.
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Dot w1, w2, w3.
10:02
Okay, so to find this, this dot product, we need to find the product of our x values, which is u1, w1, plus the product of our y values, which is plus u2, w2, plus the product of our z values, which is plus u3w3.
10:20
Okay, so that's u .s.
10:22
So we need to multiply that by this our vector v.
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So, to multiply the scalar by our vector v, which is v1, v2, b3, which is, so when we do that, we need to multiply this into every one of these.
10:49
So u .w times v is equal to v1 times u1 plus u2 w2, plus u2, and u3w3, v2 times u1w1 plus u2w2 plus u3 -w3 comma b3 times u1 u2w1 plus u2 2w2 plus u3 u3w3.
11:22
Okay, so that's this.
11:25
So now we're adding that to or subtracting u .dv times w.
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So let's find out what u .dv is.
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So u .v is going to be equal to u1, u2, u3, dotted with v, which is v1, v2, v3.
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So finding the product of our x components is u1 v1, plus the product of our y components is u2 v2, plus the product of our z components is u3 v3.
11:56
So this is u1, u .db.
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So we need to multiply this by our w.
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Vector.
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So we're multiplying this scalar quantity to our w vector, which is going to the vector w1, w2, w3.
12:13
Okay, so this becomes u .d .v times w is equal to w1, u1 v1 plus u2 v2 plus u3 v3...