Vector Algebra · JEE Main previous year questions

A vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $\vec{a}$ has components $p+1$ and $\sqrt{10}$, then the value of $p$ is equal to :

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

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}|$. If $|\vec{a}\times \vec{b}|=|\vec{a}|$, then the angle between the vectors $\left(\vec{a}+\vec{b}+\left(\vec{a}\times \vec{b}\right)\right)$ and $\vec{a}$ is equal to :

1. A.${\sin }^{-1}\left(\frac{1}{\sqrt{6}}\right)$
2. B.${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
3. C.${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
4. D.${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

In a triangle ABC, if $|\overrightarrow{BC}|=8$, $|\overrightarrow{CA}|=7$, $|\overrightarrow{AB}|=10$, then the projection of the vector $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is equal to :

1. A.$\frac{25}{4}$
2. B.$\frac{127}{20}$
3. C.$\frac{85}{14}$
4. D.$\frac{115}{16}$

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha ,\beta )$, $(0,\beta )$ and $(0,0)$ is equal to :

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

Let the position vectors of two points P and Q be $3\hat{i}-\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}-4\hat{k}$, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are $(4,-1,2)$ and $(-2,1,-2)$, respectively. Let lines PR and QS intersect at T. If the vector $\overrightarrow{TA}$ is perpendicular to both $\overrightarrow{PR}$ and $\overrightarrow{QS}$ and the length of vector $\overrightarrow{TA}$ is $\sqrt{5}$ units, then the modulus of a position vector of A is :

1. A.$\sqrt{171}$
2. B.$\sqrt{227}$
3. C.$\sqrt{482}$
4. D.$\sqrt{5}$

If $\vec{a}$ and $\vec{b}$ are perpendicular, then $\vec{a}\times \left(\vec{a}\times \left(\vec{a}\times \left(\vec{a}\times \vec{b}\right)\right)\right)$ is equal to :

1. A.$\frac{1}{2}|\vec{a}{|}^4\vec{b}$
2. B.$\vec{0}$
3. C.$\vec{a}\times \vec{b}$
4. D.$|\vec{a}{|}^4\vec{b}$

If $(1,5,35)$, $(7,5,5)$, $(1,\lambda ,7)$ and $(2\lambda ,1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is :

1. A.$-\frac{44}{5}$
2. B.$-\frac{39}{5}$
3. C.$\frac{44}{5}$
4. D.$\frac{39}{5}$

If vectors $\overrightarrow{a_1}=x\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{a_2}=\hat{i}+y\hat{j}+z\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i}+y\hat{j}+z\hat{k}$ is :

1. A.$\frac{1}{\sqrt{3}}\left(\hat{i}-\hat{j}+\hat{k}\right)$
2. B.$\frac{1}{\sqrt{2}}\left(-\hat{j}+\hat{k}\right)$
3. C.$\frac{1}{\sqrt{2}}\left(\hat{i}-\hat{j}\right)$
4. D.$\frac{1}{\sqrt{3}}\left(\hat{i}+\hat{j}-\hat{k}\right)$

Let $\vec{a}=\hat{i}+\alpha \hat{j}+3\hat{k}$ and $\vec{b}=3\hat{i}-\alpha \hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8\sqrt{3}$ square units, then $\vec{a}\cdot \vec{b}$ is equal to ______ .

If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $$\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$$ is ______ .

If $\vec{x}$ and $\vec{y}$ be two non-zero vectors such that $|\vec{x}+\vec{y}|=|\vec{x}|$ and $2\vec{x}+\lambda \vec{y}$ is perpendicular to $\vec{y}$, then the value of $\lambda$ is ______ .

The lines $\vec{r}=(\hat{i}-\hat{j})+l(2\hat{i}+\hat{k})$ and $\vec{r}=(2\hat{i}-\hat{j})+m(\hat{i}+\hat{j}+\hat{k})$

1. A.do not intersect for any values of $l$ and m
2. B.intersect for all values of $l$ and m
3. C.intersect when $l=2$ and $\mathrm{m}=\frac{1}{2}$
4. D.intersect when $l=1$ and $\mathrm{m}=2$

Let $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right],x\in \mathbb{R}$ and $A^4=[a_{ij}]$. If $a_{11}=109$, then $a_{22}$ is equal to _______.

Let $a,b,c\in \mathbb{R}$ be such that $a^2+b^2+c^2=1$. If $a\cos \theta =b\cos \left(\theta +\frac{2\pi }{3}\right)=c\cos \left(\theta +\frac{4\pi }{3}\right)$, where $\theta =\frac{\pi }{9}$, then the angle between the vectors $a\hat{i}+b\hat{j}+c\hat{k}$ and $b\hat{i}+c\hat{j}+a\hat{k}$ is :

1. A.0
2. B.$\frac{\pi }{9}$
3. C.$\frac{2\pi }{3}$
4. D.$\frac{\pi }{2}$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that ${\left|\vec{a}-\vec{b}\right|}^2+{\left|\vec{a}-\vec{c}\right|}^2=8$. Then ${\left|\vec{a}+2\vec{b}\right|}^2+{\left|\vec{a}+2\vec{c}\right|}^2$ is equal to _______.

Let the position vectors of points 'A' and 'B' be $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k}$, respectively. A point 'P' divides the line segment AB internally in the ratio $\lambda :1(\lambda >0)$. If O is the origin and ${\left|\overrightarrow{OB}\cdot \overrightarrow{OP}-3\left|\overrightarrow{OA}\times \overrightarrow{OP}\right|\right|}^2=6$, then $\lambda$ is equal to _______.

The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points (–1, 2, 3) and (3, –2, 10) is ____________.

If the vectors, $\vec{p}=(a+1)\hat{i}+a\hat{j}+a\hat{k}$, $\vec{q}=a\hat{i}+(a+1)\hat{j}+a\hat{k}$ and $\vec{r}=a\hat{i}+a\hat{j}+(a+1)\hat{k}(a\in \mathbb{R})$ are coplanar and $3(\vec{p}\cdot \vec{q}{)}^2-\lambda |\vec{r}\times \vec{q}{|}^2=0$, then the value of $\lambda$ is _______.

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{3}$, $|\vec{b}|=5$, $\vec{b}\cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi }{3}$. If $\vec{a}$ is perpendicular to the vector $\vec{b}\times \vec{c}$, then $\left|\vec{a}\times \left(\vec{b}\times \vec{c}\right)\right|$ is equal to _______.

Let $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b}\times \vec{c}=\vec{b}\times \vec{a}$ and $\vec{c}\cdot \vec{a}=0$, then $\vec{c}\cdot \vec{b}$ is equal to

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

A vector $\vec{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}$ $(\alpha ,\beta \in \mathbb{R})$ lies in the plane of the vectors, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4\hat{k}$. If $\vec{a}$ bisects the angle between $\vec{b}$ and $\vec{c}$, then:

1. A.$\vec{a}\cdot \hat{i}+3=0$
2. B.$\vec{a}\cdot \hat{k}-4=0$
3. C.$\vec{a}\cdot \hat{i}+1=0$
4. D.$\vec{a}\cdot \hat{k}+2=0$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. If $\lambda =\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ and $\vec{d}=\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}$, then the ordered pair, $(\lambda ,\vec{d})$ is equal to :

1. A.$\left(\frac{3}{2},3\vec{a}\times \vec{c}\right)$
2. B.$\left(-\frac{3}{2},3\vec{c}\times \vec{b}\right)$
3. C.$\left(-\frac{3}{2},3\vec{a}\times \vec{b}\right)$
4. D.$\left(\frac{3}{2},3\vec{b}\times \vec{c}\right)$

Let $\vec{a}=3\hat{i}+2\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-2\hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ has the magnitude 12 then one such vector is :

1. A.$4\left(2\hat{i}-2\hat{j}-\hat{k}\right)$
2. B.$4\left(-2\hat{i}-2\hat{j}+\hat{k}\right)$
3. C.$4\left(2\hat{i}+2\hat{j}+\hat{k}\right)$
4. D.$4\left(2\hat{i}+2\hat{j}-\hat{k}\right)$

If the volume of parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k}$, $\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum, then $\lambda$ is equal to :

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

Let $\alpha \in \mathbb{R}$ and the three vectors $\vec{a}=\alpha \hat{i}+\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\alpha \hat{k}$ and $\vec{c}=\alpha \hat{i}-2\hat{j}+3\hat{k}$. Then the set $S=\{\alpha :\vec{a},\vec{b}\text{ and }\vec{c}\text{ are coplanar}\}$ :

1. A.contains exactly two numbers only one of which is positive
2. B.is singleton
3. C.contains exactly two positive numbers
4. D.is empty