14 prove the followings 1 triple vector product veca times...

14. Prove the followings: (1) (triple vector product) $(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{b} \cdot \vec{c})\vec{a}$ (2) $(\vec{a} \times \vec{b}).(\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})$ (3) (Jacobi identity) $(\vec{a} \times \vec{b}) \times \vec{c} + (\vec{b} \times \vec{c}) \times \vec{a} + (\vec{c} \times \vec{a}) \times \vec{b} = \vec{0}$

Transcript

0:00 Hello there.

00:02 For this exercise we need to prove some identities related with the cross problems.

00:07 So first we are going to start with these identities of the triple vector product.

00:15 For that we are going to consider the following that these vectors are vectors that have the following components, a1, a2, a3, b as well.

00:31 B1 b2 and b3 and of course c will also be c1 c2 and c3 so we're going to start with the first identity that is this one here for that we're going to consider the left -hand side first and then we're going to compare with the right -hand side here represented with some product okay so let's start with the triple cross product given by a cross b cross c which is the left -hand side of the expression that we have before so first the cross -product of a would be we know that these can be computed using the determinant where here we put the coordinate vectors ijk and here in the corresponding rows we put the components of the vector so that means a 1 a 2 a 3 b 1 b2 and b 3 so in this case this cross -product is equal to the vector i'm going to put as a column vector because i think that is easier to visualize is a 2 b3 minus a 3 b2 b2 a 3b1 minus a 1 b3 and a 1 b2 minus a 2 b1 b1 okay so this is the corresponding cross product between a with b and now we need to take the cross product of a cross b that means the previous vector that we obtained it with c so in that case the the will be a little bit bigger.

02:42 So we have here a jeth and decay one and two.

02:56 Okay, so the first vector that we're going to put here on the first row will be the vector that we just computed.

03:04 So in the first component we have a 2b3 minus a 3 b2.

03:10 Here we have a 3 b2.

03:11 Here we have a 3 b1 minus a 1 b3.

03:17 And here in the case component we have a1 b2 minus a2b1 and the last row will be just the vector c that we just defined it previously that means here c1 c2 and c3 the result again i'm going to put as a row as a column vector because it is easier to be visualize what is happening so in this case the result after taking the determinant of this matrix is a vector i think that i'm not going to have enough space i'm going to put it below so the resulting vector is a 2 v1 c2 minus a 1 b2 b2, another color, minus a1, b2c2, plus a 3 b1 c3, and to minus a1 b3 3.

04:39 Okay, so here we have the vector obtained after taking that determinant.

04:48 So you can observe that i have two colors here to represent the positive and the negative values.

04:54 Values, okay? so this corresponds to the left -hand side of our expression, okay? and we're going to come back later to this result.

05:02 Now we need to observe what happens with the right -hand side of our expression.

05:07 So the right -hand side of our expression was the dot product of a with c times the vector b and minus the dot product of b with c times the vector a.

05:22 Okay, so we know that the product of a with c, it's going to be just the sum from i equals to 1 up to 3 of a sub i and c sub i.

05:35 Or if we write the whole expression is a1, b1 plus a2 b2 plus a3 b3.

05:49 And similarly, this dot product between b and c.

05:53 Is not more than taking b1 c1 plus b2 c2 and b3 okay so we have this dot products and we need to multiply to the corresponding vectors b and a so when we do that we obtain a big expression so i'm going to put directly here what we obtain from considering this sum of the products with vector and of the products that multiplies vectors.

06:30 Okay, so this expression is equal.

06:36 So i think that i'm going to take this first so we'll be this.

06:47 I'm going to copy three times i'm going to show you y and the same with this one.

07:05 Okay, so this resulting vector that we're going to have here, put minus here, minus here, a.

07:17 .c times the vector b will return this part and i'm going to put it in science.

07:22 So is the sum of elements of the vectors a and c, let me see, oh yeah, we forgot.

07:33 Here is c, not b.

07:37 I got a small mistake there.

07:50 So this dot product of a with c times the vector b corresponds to all that i'm going to put here in cyan.

07:58 So times the first component.

08:00 This also will multiply the second component of the vector and of course the third one.

08:06 And this part, so in cyan i'm putting this part.

08:11 The multiplication of the dot product of a with c times the vector b...

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