91Ó°ÊÓ

Let \(A B C D\) be a parallelogram such that \(\overrightarrow{A B}=\vec{q}, \overrightarrow{A D}=\bar{p}\) and \(\square B A D\) be an acute angle. If \(\vec{r}\) is the vector which coincides with the altitude directed from the vertex \(B\) to the side \(A D\), then \(\vec{r}\) is given by \([2012]\) (A) \(\vec{r}=3 \vec{q}-\frac{3(\vec{p} \cdot \vec{q})}{(\vec{p} \cdot \vec{p})} \vec{p}\) (B) \(\vec{r}=-\vec{q}+\left(\frac{\vec{p} \cdot \vec{q}}{\vec{p} \cdot \vec{p}}\right) \vec{p}\)

If the vectors \(\overrightarrow{A B}=3 \hat{i}+4 \hat{k}\) and \(\overrightarrow{A C}=5 \hat{i}+2 \hat{j}+4 \hat{k}\) represent the sides of a triangle \(A B C\), then the length of the median through \(A\) is (A) \(\sqrt{72}\) (B) \(\sqrt{33}\) (C) \(\sqrt{45}\) (D) \(\sqrt{18}\)

If \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]=\lambda[\vec{a} \vec{b} \vec{c}]^{2}\), then the value of \(\lambda\) is equal to [2014] (A) 2 (B) 3 (C) 0 (D) 1

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that no two of them are collinear and \((\vec{a} \times \vec{b}) \times \vec{c}=\frac{1}{3}|\vec{b} \| \vec{c}| \vec{a}\). If \(\theta\) is the angle between vectors \(\vec{b}\) and \(\vec{c}\), then a value of \(\sin \theta\) (A) \(\frac{-\sqrt{2}}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{-2 \sqrt{3}}{3}\) (D) \(\frac{2 \sqrt{2}}{3}\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three unit vectors such that \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\sqrt{3}}{2}(\vec{b}+\vec{c})\). if \(\vec{b}\) is not parallel to \(\vec{c}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is (A) \(\frac{5 \pi}{6}\) (B) \(\frac{3 \pi}{4}\) (C) \(\frac{\pi}{2}\) (D) \(\frac{2 \pi}{3}\)