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Vector Product. Results in a vector. Dot product (Scalar product). Results in a scalar a · b = a x b x +a y b y +a z b z. 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.
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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
