Let $ A B C D $ be a parallelogram such that $ \overrightarrow{A B}=\vec{q}, \overrightarrow{A D}=\vec{p} $ and $ \angle B A D $ be an acute angle. If $ \vec{r} $ is the vector that coincides with the altitude directed from the vertex $ B $ to the side $ A D $, then $ \vec{r} $ is given by

Question

Let $ A B C D $ be a parallelogram such that $ \overrightarrow{A B}=\vec{q}, \overrightarrow{A D}=\vec{p} $ and $ \angle B A D $ be an acute angle. If $ \vec{r} $ is the vector that coincides with the altitude directed from the vertex $ B $ to the side $ A D $, then $ \vec{r} $ is given by, $ \vec{r}=\vec{q}-\left(\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}\right) \vec{p} $, $ \vec{r}=3 \vec{q}+\frac{3(\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}+\left(\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}\right) \vec{p} $

MARKS App · Accepted Answer

By Triangle law of Vector addition r → = B A → + A Q → r → = - q → + λ p → ∵ r → is Perpendicular to the vector p → ⇒ r → · p → = 0 ⇒ - q → + λ p → · p → = - q · → p → + λ p · → p → = 0 ⇒ λ = q · → p → p · → p → ∴ r → = - q → + q · → p → p → · p → p →

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