# Vector Solutions, Shortcuts, Formulas

## Observation

• $\vec{a}\cdot \vec{b}$ is real number
• Let $\vec{a}\:and\:\vec{b},$ be two nonzero vector , then $\vec{a}\cdot \vec{b}=0$ if and only if $\vec{a}\:and\:\vec{b},$ are perpendicular to each other i.e $\vec{a}\cdot \vec{b}=0$ ,
• If θ = 0, then , $\vec{a}.\vec{b}=|\vec{a}||\vec{b}|$ in particular $\vec{a}\cdot \vec{a}=|\vec{a}^{2}|$ as θ in this case is 0.
• If θ = π , then $\vec{a}\cdot \vec{b}=-|\vec{a}||\vec{b}|$ in particular $\vec{a}\cdot \vec{a}=|\vec{a}^{2}|$ as θ in this case is $\Pi$
• In view of the Observations 2 and 3, for mutually perpendicular unit vector $\hat{i},\hat{j},\hat{k}$ we have , $\hat{i}\cdot \hat{i}=\hat{j}\cdot \hat{j}=\hat{k}\cdot \hat{k}=1\: or\: \hat{i}\cdot \hat{j}=\hat{j}\cdot \hat{k}=\hat{k}\cdot \hat{i}=0$
• The angle between two non zero vector $\vec{a}\:and\:\vec{b}$ is given by $\cos\:\:\frac{\vec{a}\:\:\vec{b}}{\left | \vec{a} \right |\left | \vec{b} \right |}\:\:or\:\:\theta$
$=\cos^{-1}\:\left ( \frac{\vec{a}\:\:\vec{b}}{\left | \vec{a} \right |\left | \vec{b} \right |} \right )$
• The scalar product is commutative i.e $\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$
• Cross product of two vector : he three dimensional right handed rectangular coordinate system. In this system, when the positive x -axis is rotated counter clockwise nto the positive y -axis, a right handed (standard) screw would advance in the direction of the positive z -axis (Fig 10.22(i)). In a right handed coordinate system, the thumb of the right hand points in the direction of the positive z -axis when the fingers are curled in the direction away from the positive x-axis toward the positive y -axis (Fig 10.22(ii).

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