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Find the cross product $ a \times b $ and verify that it is orthogonal to both $ a $ and $ b $.
$ a = 3i + 3j - 3k $ , $ b = 3i - 3j + 3k $
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$a \times b=0 i-18 j-18 k$
Vectors and the Geometry of Space
The Cross Product
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In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.
In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.
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Let's try another cross product question. We're taking a cross product of two vectors A&B. And let's write them in our Matrix A. Is three, I plus three J minus three K. And b. is three I minus three. J plus three K. And we can use the formula in the textbook. We ignore our first column and we look at three times 3 minus negative three times negative three. Let's keep track of all these minus signs. three times 3 -3 negative three times minus three I minus. And then we ignore the second column and look at the product of three times three minus negative three times three. So that's three times three minus -3 times three jay plus. And then we'll look at the third column or will ignore the third column And look at three times -3 Times three times 3, three times -3 -3 times three. There we go, That's three times -3 minus three times three. Okay, let's start to simplify this, three times three is nine And -3 times -3 is also nine. So 9 -9 I -3 times three is 9 -3 times three is -9. So we have nine minus minus nine. There's nine plus nine. Okay. Plus three times -3 is minus nine minus three times 3 is nine. Okay Simplifying this a little more, we know that 9 -9 is zero. I so we don't even need to write that if we don't want minus nine plus nine is 18 J And then -9 -9. That's another -18. Okay Or we can write that as the vector zero minus 18 minus 18. Thanks for watching.
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