Embibe Experts Solutions for Chapter: Vector Algebra, Exercise 1: JEE Advanced Paper 1 - 2018

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 $\overrightarrow{OC}=\frac{1}{2}\left(\overrightarrow{OB}-\lambda \overrightarrow{OA}\right)$ for some $\lambda >0.$ If $\left|\overrightarrow{OB}\times \overrightarrow{OC}\right|=\frac{9}{2},$ then which of the following statements is (are) TRUE?

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

Let $\overrightarrow{a}=2\widehat{i}+\widehat{j}-\widehat{k}$ and $\overrightarrow{b}=\widehat{i}+2\widehat{j}+\widehat{k}$ be two vectors. Consider a vector $\overrightarrow{c}=\alpha \overrightarrow{a}+\beta \overrightarrow{b}, \alpha , \beta \in R.$ If the projection of $\overrightarrow{c}$ on the vector $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ is $3\sqrt{2},$ then the minimum value of $\left(\overrightarrow{c}-\left(\overrightarrow{a}\times \overrightarrow{b}\right)\right).\overrightarrow{c}$ equals

Consider the cube in the first octant with sides $OP, OQ$ and $OR$ of length $1$, along the $x$-axis, $y$-axis and $z$-axis, respectively, where $O(0,0,0)$ is the origin. Let $S\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ be the centre of the cube and $T$ be the vertex of the cube opposite to the origin $O$ such that $S$ lies on the diagonal $OT$. If $\overrightarrow{p}=\overrightarrow{SP}, \overrightarrow{q}=\overrightarrow{SQ}, \overrightarrow{r}=\overrightarrow{SR}$ and $\overrightarrow{t}=\overrightarrow{ST}$, then the value of $\left|\left(\overrightarrow{p}\times \overrightarrow{q}\right)\times \left(\overrightarrow{r}\times \overrightarrow{t}\right)\right|$ is ______.

Let $\widehat{u}=u_1\widehat{i}+u_2\widehat{j}+u_3\widehat{k}$ be a unit vector in ${\mathbb{R}}^3$ and $\widehat{w}=\frac{1}{\sqrt{6}} \left(\widehat{i}+ \widehat{j}+2\widehat{k}\right).$ Given that there exists a vector $\overrightarrow{v}$ in ${\mathbb{R}}^3$ such that $\left|\widehat{u}\times \overrightarrow{v}\right|=1$ and $\widehat{w} . \left(\widehat{u}\times \overrightarrow{v}\right)=1.$ Which of the following statement(s) is (are) correct ?

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

In a triangle $PQR$ , let $\overrightarrow{a}=\overrightarrow{QR}, \overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}$ .

If $\left|\overrightarrow{a}\right|=3, \left|\overrightarrow{b}\right|=4$ and $\frac{\overrightarrow{a}.\left(\overrightarrow{c}-\overrightarrow{b}\right)}{\overrightarrow{c}.\left(\overrightarrow{a}-\overrightarrow{b}\right)}=\frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|}$, then the value of ${\left|\overrightarrow{a}\times \overrightarrow{b}\right|}^2$ is _______

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=0.$ For some $x,y\in \mathrm{R,}$ let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+\left(\overrightarrow{a}\times \overrightarrow{b}\right).$ If $\left|\overrightarrow{c}\right|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b},$ then the value of $8{\cos }^2\alpha$ is