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- https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_I_(Ellingson)/04%3A_Vector_Analysis/4.01%3A_Vector_ArithmeticIn mathematical notation, a real-valued vector A is said to have a magnitude A=|A| and direction a^ such that A=Aa^(4.1.1) where a^ is a unit vector (i.e., a real-valued vector having magnitud...In mathematical notation, a real-valued vector A is said to have a magnitude A=|A| and direction a^ such that A=Aa^(4.1.1) where a^ is a unit vector (i.e., a real-valued vector having magnitude equal to one) having the same direction as A . If a vector is complex-valued, then A is similarly complex-valued
- https://phys.libretexts.org/Bookshelves/University_Physics/Calculus-Based_Physics_(Schnick)/Volume_A%3A_Kinetics_Statics_and_Thermodynamics/21A%3A_Vectors_-_The_Cross_Product_and_TorqueDo not use your left hand when applying either the right-hand rule for the cross product of two vectors (discussed in this chapter) or the right-hand rule for “something curly something straight” disc...Do not use your left hand when applying either the right-hand rule for the cross product of two vectors (discussed in this chapter) or the right-hand rule for “something curly something straight” discussed in the preceding chapter.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/66%3A_Appendices/66.16%3A_Vector_Arithmeticwhere \(\mathbf{i}\) is a unit vector (a vector of magnitude 1) in the \(x\) direction, \(\mathbf{j}\) is a unit vector in the \(y\) direction, and \(\mathbf{k}\) is a unit vector in the \(z\) directi...where \(\mathbf{i}\) is a unit vector (a vector of magnitude 1) in the \(x\) direction, \(\mathbf{j}\) is a unit vector in the \(y\) direction, and \(\mathbf{k}\) is a unit vector in the \(z\) direction. \(A_{x}, A_{y}\), and \(A_{z}\) are called the \(x, y\), and \(z\) components (respectively) of vector \(\mathbf{A}\), and are the projections of the vector onto those axes.
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/10%3A_The_Dot_Product/10.01%3A_DefinitionThe dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) (written \(\mathbf{A} \cdot \mathbf{B}\), and pronounced "A dot \(\mathbf{B}\) ") is defined to be the product of their magnitudes, tim...The dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) (written \(\mathbf{A} \cdot \mathbf{B}\), and pronounced "A dot \(\mathbf{B}\) ") is defined to be the product of their magnitudes, times the cosine of the angle between them: It turns out that this combination occurs frequently in physics; the dot product is related to the projection of one vector onto the the other.
- https://phys.libretexts.org/Courses/Berea_College/Electromagnetics_I/04%3A_Vector_Analysis/4.01%3A_Vector_ArithmeticIn mathematical notation, a real-valued vector A is said to have a magnitude A=|A| and direction a^ such that A=Aa^(4.1.1) where a^ is a unit vector (i.e., a real-valued vector having magnitud...In mathematical notation, a real-valued vector A is said to have a magnitude A=|A| and direction a^ such that A=Aa^(4.1.1) where a^ is a unit vector (i.e., a real-valued vector having magnitude equal to one) having the same direction as A . If a vector is complex-valued, then A is similarly complex-valued
- 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/Courses/Gettysburg_College/Gettysburg_College_Physics_for_Physics_Majors/06%3A_C6)_Conservation_of_Angular_Momentum_I/6.01%3A_Angular_MomentumThe variables here are the rotational analogues of the linear quantities; \(\vec{\omega}\) is angular velocity (like linear velocity \(\vec{v}\)), and \(I\) is called the moment of inertia, and is ana...The variables here are the rotational analogues of the linear quantities; \(\vec{\omega}\) is angular velocity (like linear velocity \(\vec{v}\)), and \(I\) is called the moment of inertia, and is analogous to the mass \(m\). The right-hand rule is such that if the fingers of your right hand wrap counterclockwise from the x-axis (the direction in which \(\theta\) increases) toward the y-axis, your thumb points in the direction of the positive z-axis (Figure \(\PageIndex{4}\)).
- https://phys.libretexts.org/Courses/Prince_Georges_Community_College/General_Physics_I%3A_Classical_Mechanics/35%3A_The_Cross_ProductMany of the equations involving rotational motion of bodies involve the vector cross product, so before proceeding further, let's examine the cross product of two vectors in some detail. In the cross ...Many of the equations involving rotational motion of bodies involve the vector cross product, so before proceeding further, let's examine the cross product of two vectors in some detail. In the cross product, one multiplies a vector by another vector, and gets another vector back as the result (unlike the dot product, which returns a scalar result). Unlike the other two kinds of vector multiplication, the cross product is only defined for three-dimensional vectors. \({ }^{1}\)
- https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_I_-_Classical_Mechanics_(Gea-Banacloche)/09%3A_Rotational_Dynamics/9.02%3A_Angular_MomentumFor the general case, on the other hand, we have the situation shown in Figure \(\PageIndex{1}\): if the instantaneous velocity of the particle is \(\vec v\), and we draw the position vector of the pa...For the general case, on the other hand, we have the situation shown in Figure \(\PageIndex{1}\): if the instantaneous velocity of the particle is \(\vec v\), and we draw the position vector of the particle, \(\vec r\), with the point O as the origin, then the distance between O and the line of motion (sometimes also called the perpendicular distance between O and the particle) is given by \(r \sin \theta\), where \(\theta\) is the angle between the vectors \(\vec r\) and \(\vec v\).
- 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 \(\hat{\mathbf{e}}\).
