First slide

Dot product or scalar product of vectors

Question

If vector $\vec{a} and \vec{b}$ are two adjacent sides of a parallelogram, then the vector representing the altitude of the parallelogram which is perpendicular to $\vec{a}$ is

Moderate

A

$\vec{b} + \frac{\vec{b} \times \vec{a}}{| \vec{a} |^{2}}$
B

$\frac{\vec{a} \cdot \vec{b}}{| \vec{b} |^{2}}$
C

$\vec{b} - \frac{\vec{b} \cdot \vec{a}}{| \vec{a} |^{2}} \vec{a}$
D

$\frac{\vec{a} \times (\vec{b} \times \vec{a})}{| \vec{b} |^{2}}$

Solution

$\begin{array}{ll} Let \vec{OD} = t \vec{a} \\ ∴ \vec{DB} = \vec{b} - \vec{ta} \\ or (\vec{b} - \vec{ta}) \cdot \vec{a} = 0 (∵ \vec{DB} ⊥ \vec{OA}) \end{array}$
$\begin{array}{ll} or t = \frac{\vec{b} \cdot \vec{a}}{| \vec{a} |^{2}} \\ ∴ \vec{DB} = \vec{b} - \frac{(\vec{b} \cdot \vec{a}) \vec{a}}{| \vec{a} |^{2}} \end{array}$

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