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- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/35%3A_The_Cross_Product/35.02%3A_PropertiesA better way to remember both products in Eqs. (32.12) and (32.13) is: "The middle vector times the dot product of the two on the ends, minus the dot product of the two vectors straddling the parenthe...A better way to remember both products in Eqs. (32.12) and (32.13) is: "The middle vector times the dot product of the two on the ends, minus the dot product of the two vectors straddling the parenthesis times the remaining one.”
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/35%3A_The_Cross_Product/35.01%3A_Definition_and_FormsBy convention, we choose the one given by a right-hand rule: if you curl the fingers of your right and from vector \(\mathbf{A}\) toward vector \(\mathbf{B}\), then the thumb of your right hand points...By convention, we choose the one given by a right-hand rule: if you curl the fingers of your right and from vector \(\mathbf{A}\) toward vector \(\mathbf{B}\), then the thumb of your right hand points in the direction of \(\mathbf{A} \times \mathbf{B}\) (Fig. \(\PageIndex{1}\)). \[ =\left(A_{y} B_{z}-A_{z} B_{y}\right) \mathbf{i}-\left(A_{x} B_{z}-A_{z} B_{x}\right) \mathbf{j}+\left(A_{x} B_{y}-A_{y} B_{x}\right) \mathbf{k} .\]
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/17%3A_Two-Dimensional_Rotational_Dynamics/17.02%3A_Vector_Product_(Cross_Product)Solution: Let \(\overrightarrow{\mathbf{A}}=A_{\|} \hat{\mathbf{n}}+A_{\perp} \hat{\mathbf{e}}\) where \(A_{\|}\) is the component \(\overrightarrow{\mathbf{A}}\) in the direction of \(\hat{\mathbf{n}...Solution: Let \(\overrightarrow{\mathbf{A}}=A_{\|} \hat{\mathbf{n}}+A_{\perp} \hat{\mathbf{e}}\) where \(A_{\|}\) is the component \(\overrightarrow{\mathbf{A}}\) in the direction of \(\hat{\mathbf{n}}, \hat{\mathbf{e}}\) is the direction of the projection of \(\overrightarrow{\mathbf{A}}\) in a plane perpendicular to \(\hat{\mathbf{n}}\), and \(A_{\perp}\) is the component of \(\overrightarrow{\mathbf{A}}\) in the direction of \(\hat{\mathbf{e}}\).
- https://phys.libretexts.org/Bookshelves/Classical_Mechanics/Classical_Mechanics_(Dourmashkin)/03%3A_Vectors/3.04%3A_Vector_Product_(Cross_Product).We note that the same rule applies for the unit vectors in the \(y\) and \(z\) directions, \[\hat{\mathbf{j}} \times \hat{\mathbf{k}}=\hat{\mathbf{i}}, \quad \hat{\mathbf{k}} \times \hat{\mathbf{i}}=\...We note that the same rule applies for the unit vectors in the \(y\) and \(z\) directions, \[\hat{\mathbf{j}} \times \hat{\mathbf{k}}=\hat{\mathbf{i}}, \quad \hat{\mathbf{k}} \times \hat{\mathbf{i}}=\hat{\mathbf{j}} \nonumber \] By the anti-commutatively property (1) of the vector product, \[\hat{\mathbf{j}} \times \hat{\mathbf{i}}=-\hat{\mathbf{k}}, \quad \hat{\mathbf{i}} \times \hat{\mathbf{k}}=-\hat{\mathbf{j}} \nonumber \] The vector product of the unit vector \(\hat{\mathbf{i}}\) with itse…
