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Numerade Educator

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Problem 21 Easy Difficulty

Show that $ 0 \times a = 0 = a \times 0 $ for any vector $ a $ in $ V_3 $.

Answer

See explanation.

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Video Transcript

Mhm. Let's do another cross product problem where we're trying to show that a cross zero equals zero, no matter what factor is. This is one of the properties of cross products. But we can verify this by taking an arbitrary vector A defined to be a one A two, A three. There's going to be any numbers. And looking at what happens when we take the cross product with the zero vector 000 Using the technique from our textbook, we can ignore the first column and look at A two time 0 minus a three time zero. Hi minus next. We'll ignore our second column and look at a one time zero -3 times zero. And turn off and lastly sorry, that LBJ. And then lastly, we'll ignore our third column. Look at a one time 0 minus a two time zero, which gives us the vector zero, I plus zero J plus zero K. Which we can just call the big old zero vector. Similarly, if we want to look at what happens when we look at zero cross A. Again, we're expecting this to be zero because that's one of the properties of cross products. Let's do the same thing A one A two, A three. And if we do our math again, ignore the first column zero times a three minus zero times a two, second column, zero times a 3 0 times a one zero. And lastly zero times A to write a zero times a one just like before we get zero in every term, no matter what the values of a one, A two And a three r. Because a cross zero is equal to zero. Cross a Is equal to the zero vector. Thanks for watching.