# 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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