Let ABCD be a parallelogram such that \overset{\rightarrow}{AB} = \overset{\rightarrow}{q} \textrm{ } , \textrm{ }\textrm{ } \overset{\rightarrow}{AD} = \overset{\rightarrow}{p} and \angleBAD be an acute angle. If \overset{\rightarrow}{r} is the vector that coincides with the altitude directed from the vertex B to the side AD, then \overset{\rightarrow}{r} is given by :

Engineering

Mathematics

Scalar Product of Two Vectors

Question

Let ABCD be a parallelogram such that $\vec{AB} = \vec{q}, \vec{AD} = \vec{p}$ and $∠$ BAD be an acute angle. If $\vec{r}$ is the vector that coincides with the altitude directed from the vertex B to the side AD, then $\vec{r}$ is given by :

$\vec{r} = - 3 \vec{q} + \frac{3 (\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}$

$\vec{r} = \vec{q} - (\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}) \vec{p}$

$\vec{r} = 3 \vec{q} - \frac{3 (\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}$

$\vec{r} = - \vec{q} + (\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}) \vec{p}$

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$\vec{r} = λ \vec{p} - \vec{q}$

$\vec{r} \cdot \vec{p} = (λ \vec{p} - \vec{q}) \cdot \vec{p}$

$0 = λ \vec{p} \cdot \vec{p} - \vec{p} \cdot \vec{q}$

$λ = \frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}$

$∴ \vec{r} = - \vec{q} + (\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}) \vec{p}$

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