Vector Solutions, Shortcuts, Formulas

Definition

The vector product of two nonzero vectors \vec{a}\:and\:\vec{b} , is denoted by \vec{a}\times \vec{b}=|\vec{b}||\vec{a}|\sin\theta \hat{n} where, θ is the angle between \vec{a}\:and\:\vec{b} and ≤θ≤π and \hat{n} is a unit vector perpendicular to both \vec{a}\:and\:\vec{b} , such that , \vec{a}\:,\:\vec{b}\:and\: \hat{n} and form a right handed system (Fig 10.23). i.e., the right handed system rotated from \vec{a}to\vec{b} moves in the direction of . If either , then θ is not defined and in this case, we define of \hat{n}


Observation

  • \vec{a}\times \vec{b} is a vector
  • Let \vec{a}\:and\:\vec{b} be two nonzero vector ,Then \vec{a}\times \vec{b}=\vec{0} if and only if \vec{a}\:and\:\vec{b} are parallel (or collinear) to each other . in particular \vec{a}\times \vec{b}=\vec{0}\: and\: \vec{a}\times (-\vec{a}) , since in the first situation, θ = 0 and in the second one, θ = π , making the value of sinθ to be 0
  • In view of the Observations 2 and 3, for mutually perpendicular \hat{i},\hat{k},\hat{k} (fig 10.24)
  • we have , \hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times \hat{k}=\hat{0} unit vector \hat{i},\hat{k},\hat{k}
    \hat{i}\times \hat{j}=\hat{k},\hat{j}\times \hat{k}=\hat{k},\hat{k}\times \hat{i}=\hat{j}
  • In terms of vector product, the angle between two vectors and may be given as \vec{a}\:and\:\vec{b}
  • It is always true that the vector product is not commutative, as \vec{a}\times \vec{b}=-\vec{b}\times \vec{a} Indeed, , where \vec{a}\times \vec{b}=|\vec{a}||\vec{b}|\sin \theta\hat{n} form a right handed system, i.e., θ is traversed from . \vec{a}\:to\: \vec{b} is a vector Fig 10.25 (i). While, \vec{b}\times \vec{a}=|\vec{a}||\vec{b}|\sin \theta \vec{n_{1}} ,where, \vec{b},\vec{a} ,and n_{1} form a right handed system i.e. θ is traversed from .

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