Show that $ (a \times b) \cdot b = 0 $ for all vectors $ a $ and $ b $ in $ V_3 $.
See proof for answer.
Welcome back to another cross product problem where we're going to try and find the value of a cross B dot be for any vectors, A and B. So A can be any value, can be can be any value. Now, there's a couple different ways we can try this. The naive way would be to plug in a one, A two, A three, B one B two In B three in our matrix. And then use the method in the textbook, calculating the cross product of each combination of these and then take that vector and doubt it with B. But instead, let's use properties of cross products that tells us a Crosby dot another vector. It's the same thing as a God be cross that other vector. This problem is a lot easier to figure out because B Crosby is a vector cross product itself. And we know that any time we have a vector cross product itself, or a vector cross product, something parallel to itself, That's just zero. Meaning this problem reduces to Calculating a.0 or a one a two a three dot product with 000 Which is just zero times a one Plus zero times a two Plus zero times a three for just zero. If we did all the math, the slow way would give us the exact same result, but it would take a lot longer to get better. Even your time. Thanks for watching.