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Find the vector, not with determinants, but by using properties of cross products.

$ (i \times j) \times k $

$(\mathbf{i} \times \mathbf{j}) \times \mathbf{k}=\mathbf{0}$

01:45

Wen Z.

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

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All right, welcome back. We're going to be trying another cross product problem, but this time we're not going to do the math to work out every single little bit of this. We're going to use properties of cross products to find the answer to I cross J. Cross K. Which I want to point out is not equal to I cross jay. Cross K. You can try that one for yourself, but we're going to be worrying about this problem on the left. So remember I is the vector 100 And Jay is the Vector 0, 10. We could use our usual method with a matrix in order to determine this cross product, but that's not what we're going to do. We know that I is just this factor here And has length one in J. Is just this vector here, And also has length one. And the right hand rule tells us that I cross J. As we curl our fingers from the direction of I towards the direction of J. Then the cross product will be pointing in the direction that our thumb is pointing. So as we curl our fingers from I to J on our right hand, that gives our answer pointing straight up. This is I cross J. And since both I and J had length one, I cross J is just 001 it's just K. And so this whole problem simplifies to Okay, cross, okay, another thing we know about Cross products is if we're looking at let's just do this on the side here. If we're looking at a cross B, Then this is always equal to zero whenever A is parallel to be and vice versa. And so since K is clearly parallel to K, K, cross K has to be zero, meaning this entire solution I cross J crossed with K Has to be zero. Thanks for watching.

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