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## Vector Product

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**Vector Product**Results in a vector**Dot product (Scalar product)**• Results in a scalar a · b = axbx+ayby+azbz Scalar**Vector Product**• Results in a vector =**Properties…**• a x b = - b x a • a x a = 0 • a x b = 0 if a and b are parallel. • a x (b + c) = a x b + a x c • a x(λb)= λ a x b**Examples…**• i x j = k, j x k = i, k x i = j • j x i = - k, i x k = -j, k x j = i • i x i = 0**c**a b θ Vector product c = a x b • c is perpendicular to a and b, in the direction according to the right-handed rule.**c**c a a b b Vector product – Direction: right-hand rule c = a x b θ**c**a b Vector product – right-hand rule c = a x b c a θ b**Vector product – right-hand rule**c = a x b c c b b θ a a**Vector Product-magnitude**a = (a1, 0, 0) b = (b1, b2, 0) c a x θ b2 b y**Invariance of axb**• The direction of axb is decided according the right-hand rule. • The magnitude of axb is decided by the magnitudes of a and b and the angle between a and b. axb is invariant with respect to changes from one right-handed set of axes to another.**Moment of a force about a point**M = | F |d F M = | F | |R |sinθ O R M = RxF θ d**Component of a vector a in an arbitrary direction s**a s as --- Unit vector in the direction of s ax x**Example--Component of a Force F in an arbitrary direction s**F s Fs --- Unit vector in the direction of s**Example--Component of a Moment M in an arbitrary direction s**M s Fs --- Unit vector in the direction of s ---- Scalar Triple product**Scalar Triple Product**Scalar**Volume of a parallelepiped**= Volume of the parallelepiped. F E G bxc H a θ θ b D α C c A B**Moment of a force about an axis**A F s ---- Moment of F about axis AA’ --- Unit vector in the direction of s A’**Vector Triple Product**Vector