List of top Mathematics Questions on Vectors asked in JEE Advanced

Let l₁ and l₂ be the lines r₁ = λ($\hat{i}+\hat{j}+\hat{k}$) and r₂ = ($\hat{j}-\hat{k}$) + μ ($\hat{i}+\hat{k}$), respectively. Let X be the set of all the planes H containing line l₁. For a plane H, let d (H) denote the smallest possible distance between the points of l₂ and H. Let H₀ be a plane in X for which d (H₀) is the maximum value of d (H ) as H varies over all planes in X . Match each entry in List-I to the correct entries in List-II.

 The correct option is: 

Let $\vec{u}, \vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and
$\vec{ u } \cdot \vec{ w }=1, \vec{ v } \cdot \vec{ w }=1, \vec{ w } \cdot \vec{ w }=4$
If the volume of the parallelepiped, whose adjacent sides are represented by the vectors $\vec{ u }, \vec{ v }$ and $\vec{ w }$ is $\sqrt{2}$ , then the value of $|3 \vec{u}+5 \vec{v}|$ is ______

Let $O$ be the origin and $\overrightarrow{ OA }=2 \hat{ i }+2 \hat{ j }+\hat{ k }, \overrightarrow{ OB }=\hat{ i }-2 \hat{ j }+2 \hat{ k }$ and $\overrightarrow{ OC }=\frac{1}{2}(\overrightarrow{ OB }-\lambda \overrightarrow{ OA })$ for some $\lambda $>$ 0$. If$|\overrightarrow{ OB } \times \overrightarrow{ OC }|=\frac{9}{2}$, then which of the following statements is(are) TRUE?

Let $a$ and $b$ be positive real numbers. Suppose $\overrightarrow{P Q}=a \hat{i}+b \hat{j}$ and $\overrightarrow{P S}=a \hat{i}-b \hat{j}$ are adjacent sides of a parallelogram $PQRS$. Let $\vec{ u }$ and $\vec{ v }$ be the projection vectors of $\vec{ w }=\hat{ i }+\hat{ j }$ along $\overrightarrow{ PQ }$ and $\overrightarrow{ PS }$, respectively. If $|\vec{u}|+|\vec{v}|=|\vec{w}|$ and if the area of the parallelogram $PQRS$ is $8$, then which of the following statements is/are TRUE?

Let $L_1$ and $L_2$ denote the lines $\vec{r}=\hat{i}+\lambda\left(-\hat{i}+2\hat{j}+2\hat{k}\right), \lambda\,\in\,\mathbb{R}$ and $\vec{r}=\mu\left(2\hat{i}-\hat{j}+2\hat{k}\right), \mu\,\in\,\mathbb{R}$ respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?

The edges of a parallelopiped are of unit length and are parallel to non-coplanar unit vector $\widehat{a},\widehat{b},\widehat{c}$ such that $\widehat{a}.\widehat{b}=\widehat{b}.\widehat{c}=\widehat{c}.\widehat{a}=\frac{1}{2}.$Then, the volume of the parallelopiped is