Mathematics

Problem

Let

ABCD

be a parallelogram such that

AB = \mathbf{q}

AD = \mathbf{p}

and

\angle BAD

be an acute angle. If

\mathbf{r}

is the vector that coincides with the altitude directed from the vertex

B

to the side

AD

, then

\mathbf{r}

is given by [AIEEE 2012] a.

\mathbf{r} = 3\mathbf{q} + \frac{3(\mathbf{p} \cdot \mathbf{q})}{(\mathbf{p} \cdot \mathbf{p})} \mathbf{p}

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

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

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