Explains the geometric meaning of 2D and 3D cross products, how determinants compute them, and how orientation determines their sign or direction.
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In two dimensions, the cross product of two vectors is a signed number representing the area of the parallelogram they span. Its sign depends on orientation: reversing the order of the vectors reverses the sign. The determinant of a matrix with the vectors as columns gives this signed area. This connects the calculation to linear transformations, since the determinant measures how a transformation changes area and whether it flips orientation. The area grows when the vectors become closer to perpendicular, and scaling either vector scales the cross product by the same factor.
The three-dimensional cross product instead produces a vector. Its length equals the area of the parallelogram spanned by the input vectors, and it points perpendicular to their plane. The right-hand rule determines which of the two possible perpendicular directions it takes. A determinant-based procedure computes this vector by placing the standard basis vectors in the first column and the input coordinates in the remaining columns; the resulting linear combination gives the cross product. The lecture notes that this procedure has a deeper geometric explanation involving duality, reserved for a follow-up video.
Correct answer: A. The signed area depends on the order and orientation of the vectors; with v to the right of w, v × w is positive.
Correct answer: C. Using the vectors as matrix columns gives the determinant 2·4 − 3·(−1) = 11.
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