Vector Algebra · JEE Main previous year questions

Let $\vec{a}=\alpha \hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=-2\hat{i}+\alpha \hat{j}+\hat{k}$, where $\alpha \in \mathbb{R}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $\sqrt{15({\alpha }^2+4)}$, then the value of $2{\left|\vec{a}\right|}^2+\left(\vec{a}\cdot \vec{b}\right){\left|\vec{b}\right|}^2$ is equal to :

1. A.$10$
2. B.$7$
3. C.$9$
4. D.$14$

Let $\vec{a}$ be a vector which is perpendicular to the vector $3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}$. If $\vec{a}\times (2\hat{i}+\hat{k})=2\hat{i}-13\hat{j}-4\hat{k}$, then the projection of the vector $\vec{a}$ on the vector $2\hat{i}+2\hat{j}+\hat{k}$ is :

1. A.$\frac{1}{3}$
2. B.$1$
3. C.$\frac{5}{3}$
4. D.$\frac{7}{3}$

Let $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+2\hat{k}$. Then the number of vectors $\vec{b}$ such that $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{b}|\in \{1,2,.......,10\}$ is :

1. A.0
2. B.1
3. C.2
4. D.3

Let $A_1=\{(x,y):|x|\le y^2,|x|+2y\le 8\}$ and $A_2=\{(x,y):|x|+|y|\le k\}$. If $27(\text{Area }{A}_1)=5(\text{Area }{A}_2)$, then $k$ is equal to :

Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonals of a parallelogram having area $2\sqrt{2}$. Let the angle between $\vec{a}$ and $\vec{b}$ be acute, $|\vec{a}|=1$, and $|\vec{a}\cdot \vec{b}|=|\vec{a}\times \vec{b}|$. If $\vec{c}=2\sqrt{2}\left(\vec{a}\times \vec{b}\right)-2\vec{b}$, then an angle between $\vec{b}$ and $\vec{c}$ is :

1. A.$\frac{\pi }{4}$
2. B.$-\frac{\pi }{4}$
3. C.$\frac{5\pi }{6}$
4. D.$\frac{3\pi }{4}$

Let $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=2\hat{i}-3\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{2}{\sqrt{3}}$. If $\vec{v}\cdot \hat{j}=7$, then $\vec{v}\cdot (\hat{i}+\hat{k})$ is equal to :

1. A.6
2. B.7
3. C.8
4. D.9

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ $a_i>0$, $i=1,2,3$ be a vector which makes equal angles with the coordinate axes $\mathrm{OX}$, $\mathrm{OY}$ and $\mathrm{OZ}$. Also, let the projection of $\vec{a}$ on the vector $3\hat{i}+4\hat{j}$ be $7$. Let $\vec{b}$ be a vector obtained by rotating $\vec{a}$ with $90^{\circ }$. If $\vec{a},\vec{b}$ and x-axis are coplanar, then projection of a vector $\vec{b}$ on $3\hat{i}+4\hat{j}$ is equal to:

1. A.$\sqrt{7}$
2. B.$\sqrt{2}$
3. C.$2$
4. D.$7$

Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$, where $|\vec{a}|=4$, $|\vec{b}|=3$ and $\theta \in \left(\frac{\pi }{4},\frac{\pi }{3}\right)$. Then ${\left|(\vec{a}-\vec{b})\times (\vec{a}+\vec{b})\right|}^2+4{\left(\vec{a}\cdot \vec{b}\right)}^2$ is equal to $\underline{}$.

Let $\vec{b}=\hat{i}+\hat{j}+\lambda \hat{k}$, $\lambda \in \mathbb{R}$. If $\vec{a}$ is a vector such that $\vec{a}\times \vec{b}=13\hat{i}-\hat{j}-4\hat{k}$ and $\vec{a}\cdot \vec{b}+21=0$, then $(\vec{b}-\vec{a})\cdot (\hat{k}-\hat{j})+(\vec{b}+\vec{a})\cdot (\hat{i}-\hat{k})$ is equal to $\underline{}$.

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a}\times \hat{b})|=2$. If $\theta \in (0,\pi )$ is the angle between $\hat{a}$ and $\hat{b}$, then among the statements : (S1) : $2|\hat{a}\times \hat{b}|=|\hat{a}-\hat{b}|$ (S2) : The projection of $\hat{a}$ on $(\hat{a}+\hat{b})$ is $\frac{1}{2}$

1. A.Only (S1) is true.
2. B.Only (S2) is true.
3. C.Both (S1) and (S2) are true.
4. D.Both (S1) and (S2) are false.

Let the points of intersections of the lines $x-y+1=0$, $x-2y+3=0$ and $2x-5y+11=0$ are the mid points of the sides of a triangle $\Delta \mathrm{ABC}$. Then, the area of the $\Delta \mathrm{ABC}$ is ___________.

Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\vec{a}$ is 19 units, then $|2\vec{v}{|}^2$ is equal to ___________.

Let $[t]$ denote the greatest integer less than or equal to t. Let $f(x)=x-[x]$, $g(x)=1-x+[x]$, and $h(x)=\min \{f(x),g(x)\}$, $x\in [-2,2]$. Then h is :

1. A.continuous in $[-2,2]$ but not differentiable at more than four points in $(-2,2)$
2. B.not continuous at exactly three points in $[-2,2]$
3. C.continuous in $[-2,2]$ but not differentiable at exactly three points in $(-2,2)$
4. D.not continuous at exactly four points in $[-2,2]$

1. A.$5$
2. B.$2\sqrt{10}$
3. C.$5\sqrt{3}$
4. D.$5\sqrt{2}$

If the projection of the vector $\hat{i}+2\hat{j}+\hat{k}$ on the sum of the two vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $-\lambda \hat{i}+2\hat{j}+3\hat{k}$ is 1, then $\lambda$ is equal to ________.

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b}\times (\vec{b}\times \vec{c})$. If magnitudes of the vectors $\vec{a},\vec{b}$ and $\vec{c}$ are $\sqrt{2}$, 1 and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta \left(0<\theta <\frac{\pi }{2}\right)$, then the value of $1+\tan \theta$ is equal to :

1. A.$\sqrt{3}+1$
2. B.2
3. C.1
4. D.$\frac{\sqrt{3}+1}{\sqrt{3}}$

Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k},\vec{b}=3\hat{i}+\beta \hat{j}-\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a}\cdot \vec{b}=-1$ and $\vec{b}\cdot \vec{c}=10$, then $\left(\vec{a}\times \vec{b}\right)\cdot \vec{c}$ is equal to ____.

Let the vectors $$(2+a+b)\hat{i}+(a+2b+c)\hat{j}-(b+c)\hat{k},(1+b)\hat{i}+2b\hat{j}-b\hat{k}\text{ and }(2+b)\hat{i}+2b\hat{j}+(1-b)\hat{k},a,b,c\in R$$ be co-planar. Then which of the following is true?

1. A.2b = a + c
2. B.3c = a + b
3. C.a = b + 2c
4. D.2a = b + c

Let $\vec{p}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to each of the vectors $((\vec{p}+\vec{q})$ and $(\vec{p}-\vec{q})$, and $\left|\vec{r}\right|=\sqrt{3}$, then $|\alpha |+|\beta |+|\gamma |$ is equal to ____.

Let a, b and c be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are co-planar, then c is equal to :

1. A.$\frac{2}{\frac{1}{a}+\frac{1}{b}}$
2. B.$\frac{a+b}{2}$
3. C.$\frac{1}{a}+\frac{1}{b}$
4. D.$\sqrt{ab}$

Let a vector $\vec{a}$ be coplanar with vectors $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to $\vec{d}=3\hat{i}+2\hat{j}+6\hat{k}$, and $|\vec{a}|=\sqrt{10}$. Then a possible value of $[\vec{a}\vec{b}\vec{c}]+[\vec{a}\vec{b}\vec{d}]+[\vec{a}\vec{c}\vec{d}]$ is equal to :

1. A.$-42$
2. B.$-40$
3. C.$-29$
4. D.$-38$

Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{a}\times \vec{b}=\vec{c}$, $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{a}|=2$. Then which one of the following is not true?

1. A.$\vec{a}\times ((\vec{b}+\vec{c})\times (\vec{b}-\vec{c}))=\vec{0}$
2. B.$\text{Projection of }\vec{a}\text{ on }(\vec{b}\times \vec{c})\text{ is }2$
3. C.$[\vec{a}\vec{b}\vec{c}]+[\vec{c}\vec{a}\vec{b}]=8$
4. D.$|3\vec{a}+\vec{b}-2\vec{c}{|}^2=51$

Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=|\vec{c}|$, $|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $(\vec{a}\times \vec{b})$ and $\vec{c}$ is $\frac{\pi }{6}$, then the value of $\left|(\vec{a}\times \vec{b})\times \vec{c}\right|$ is :

1. A.$\frac{2}{3}$
2. B.$4$
3. C.$3$
4. D.$\frac{3}{2}$

Let $\vec{a},\vec{b},\vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36{\cos }^{2}2\theta$ is equal to ________.

Let P be a plane passing through the points $(1,0,1)$, $(1,-2,1)$ and $(0,1,-2)$. Let a vector $\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ be such that $\vec{a}$ is parallel to the plane P, perpendicular to $(\hat{i}+2\hat{j}+3\hat{k})$ and $\vec{a}\cdot (\hat{i}+\hat{j}+2\hat{k})=2$, then $(\alpha -\beta +\gamma {)}^2$ equals ________.