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Problem 6: Find the scalar triple product & (v X w) for(a)2 U = 1 32U = 3 4W =(b)3 2 u = , U = W = 5 2| 3...

Question

Problem 6: Find the scalar triple product & (v X w) for(a)2 U = 1 32U = 3 4W =(b)3 2 u = , U = W = 5 2| 3

Problem 6: Find the scalar triple product & (v X w) for (a) 2 U = 1 3 2 U = 3 4 W = (b) 3 2 u = , U = W = 5 2| 3



Answers

Compute the scalar triple product $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})$. $$\mathbf{u}=(-2,0,6), \mathbf{v}=(1,-3,1), \mathbf{w}=(-5,-1,1)$$

And this problem, we're going to use the concept of scalar triple product in order to determine the scalar triple product of three given vectors. Now the scalar triple product that we need to determine is new dot we cross W. Now this scalar triple product is given as a determinant of order three. Where the first role has the components of the vector U, which will be minus 1 to 4. The second rule will have the components of the vector V. So that will be 34 minus two. And the third rule, we'll have the components of the vector W, which will be minus 1 to 5. Let us expand this determinant along the first row, so that will be equals two minus one into the minor of this term, which will be the determinant for minus 2 to 5 minus two times the minor of two, which will be the determinant, three minus two minus one. Fight. The minor is obtained by deleting the first row in the second column, which is where the element to lies. And with this we add four times the minor of four which will be obtained by deleting the first row and the third column of the determinant. So that would be 34 minus 12 So this will be equals two minus one times four times five which is 20 minus minus two times two which is minus four minus two times three times five which is 15 minus minus two times minus one, which is to plus four times three times two which is six minus four times minus one, which is minus four. So this is equals two minus one times 20 minus of minus four which is 20 plus four or 24 minus two times 15 minus two, which is 13 plus four times six minus of minus four which is six plus four, which is equals to 10. So this is equal to minus 24 minus 26 plus 40 and that is equals two minus depth. So the required scalar to put product in this case is equals two minus 10.

In this problem, we are going to compute the scalar triple product of three given vectors. Now the scalar triple product that we need to determine is U dot v cross double. Now this scalar triple product is given By a determinant of order three. The first row being the components of the victory you which will be E00. The second role will have the components of the vector V. There will be 00 and the third row will have the components of the vector W. So that would be 00 C. So all we need to do is determine the barrel. This determinant. So for that let us expand along the first row. So this will be eight times its minor, which is B 00 C minus zero times. It's minor, which will be zero plus zero times it's minor which will also be equal to zero. So this will just be eight times B00 c. And the value of this determinant is B times C zero times 0- zero. So all we are left with is A times B C, which means abc. So the required scalar triple product is abc.

In this problem, we're going to compute the scalar triple product of three given victims. Now the scalar triple product that we need to determine is u dot v cross Dublin. Now, in this problem, U V and w are given to be the standard unit vectors. You is given to be the standard unit victor high V is given to be the standard unit victor G, and W is given to be the standard unit victor key. Now it is a known result that D cross key is equals to the standard union victor I. So this is equals to the standard unit victor I dot the standard union victor I. Now it is a known result that I don't I is always equals to one. Any standard unit victor. The dot product of any standard unit vector with itself is equal to one. So the value of this will be equals to one. Hence the required scalar triple product. You don't be cross W. In this case is equals to one.

We can use the algebraic properties of inner product to computer values for part A first, according to the activity property. It gives us the expression in a second and then they use the cemetery here and then we can substitute the expression with their values with vehicles six times two plus four times minus six minus three times minus three minus two times seven square. Which Nicole's 12 miners, 24 plus nine miners, 98 Pacheco's miners. Hand bird. And what? And a seminar for part B. The norm of U plus V. Is the square root of the inner product of you. Plus we And it is here. Yeah. Then we use the property of relativity. It gave us this result anna's in by the cemetery. We can combine the similar terms. Have you got this expression? And we substitute the expression with their values which is the square root of one squared plus two times two plus two squared. Which he calls the square root of nine returns free.


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