Vector Algebra · JEE Main previous year questions

Let $\vec{a}=4\hat{i}-\hat{j}+3\hat{k}$, $\vec{b}=10\hat{i}+2\hat{j}-\hat{k}$ and a vector $\vec{c}$ be such that $2\left(\vec{a}\times \vec{b}\right)+3\left(\vec{b}\times \vec{c}\right)=\vec{0}$. If $\vec{a}\cdot \vec{c}=15$, then $\vec{c}\cdot (\hat{i}+\hat{j}-3\hat{k})$ is equal to :

1. A.$-6$
2. B.$-5$
3. C.$-4$
4. D.$-3$

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{a}\times \vec{c}=\vec{b}$ and $\vec{a}\cdot \vec{c}=3$, then $\vec{c}\cdot (\vec{a}-2\vec{b})$ is equal to ____.

Let $\vec{a}=2\hat{i}+3\hat{j}+3\hat{k}$ and $\vec{b}=6\hat{i}+3\hat{j}+3\hat{k}$. Then the square of the area of the triangle with adjacent sides determined by the vectors $(2\vec{a}+3\vec{b})$ and $(\vec{a}-\vec{b})$ is :

1. A.$450$
2. B.$900$
3. C.$1800$
4. D.$2400$

The square of the distance of the point $\mathrm{P}(5,6,7)$ from the line $\frac{x-2}{2}=\frac{y-5}{3}=\frac{z-2}{4}$ is equal to:

1. A.3
2. B.5
3. C.6
4. D.8

Let $\vec{a}=\sqrt{7}\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+2\hat{k}$. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}+\vec{a}\times \vec{b}=\vec{0}$ and $\vec{r}\cdot \vec{a}=0$, then $|3\vec{r}{|}^2$ is equal to:

1. A.44
2. B.54
3. C.86
4. D.132

The square of the distance of the point of intersection of the lines $\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (a\hat{i}-\hat{j})$, $a\ne 0$ and $\vec{r}=(4\hat{i}-\hat{k})+\mu (2\hat{i}+a\hat{k})$ from the origin is:

1. A.5
2. B.10
3. C.17
4. D.26

Let $\hat{u}$ and $\hat{v}$ be unit vectors inclined at an acute angle such that $|\hat{u}\times \hat{v}|=\frac{\sqrt{3}}{2}$. If $\vec{A}=\lambda \hat{u}+\hat{v}+(\hat{u}\times \hat{v})$, then $\lambda$ is equal to:

1. A.$\frac{4}{3}(\vec{A}\cdot \hat{u})-\frac{2}{3}(\vec{A}\cdot \hat{v})$
2. B.$\frac{2}{3}(\vec{A}\cdot \hat{u})-\frac{1}{3}(\vec{A}\cdot \hat{v})$
3. C.$\frac{4}{3}(\vec{A}\cdot \hat{u})+\frac{2}{3}(\vec{A}\cdot \hat{v})$
4. D.$(\vec{A}\cdot \hat{u})-\frac{1}{2}(\vec{A}\cdot \hat{v})$

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=2$ and $|\vec{b}|=3$, then the maximum value of $3\left|\left(3\vec{a}+2\vec{b}\right)\right|+4\left|\left(3\vec{a}-2\vec{b}\right)\right|$ is :

1. A.30
2. B.36
3. C.60
4. D.72

Let a line $\mathrm{L}$ passing through the point $(1,1,1)$ be perpendicular to both the vectors $2\hat{i}+2\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+2\hat{k}$. If $\mathrm{P}(a,b,c)$ is the foot of perpendicular from the origin on the line $\mathrm{L}$, then the value of $34(a+b+c)$ is :

1. A.50
2. B.80
3. C.100
4. D.120

Let the vectors $\vec{a}=-\hat{i}+\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+3\hat{j}+\hat{k}$. For some $\lambda ,\mu \in \mathbb{R}$, let $\vec{c}=\lambda \vec{a}+\mu \vec{b}$. If $\vec{c}\cdot (3\hat{i}-6\hat{j}+2\hat{k})=10$ and $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=-2$, then $|\vec{c}{|}^2$ is equal to :

1. A.8
2. B.12
3. C.14
4. D.15

Let the point A be the foot of perpendicular drawn from the point $\mathrm{P}(a,b,0)$ on the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-\alpha }{3}$. If the midpoint of the line segment PA is $\left(0,\frac{3}{4},\frac{-1}{4}\right)$, then the value of $a^2+b^2+{\alpha }^2$ is equal to :

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

For three unit vectors $\vec{a},\vec{b},\vec{c}$ satisfying $|\vec{a}-\vec{b}{|}^2+|\vec{b}-\vec{c}{|}^2+|\vec{c}-\vec{a}{|}^2=9$ and $|2\vec{a}+k\vec{b}+k\vec{c}|=3$, the positive value of k is :

1. A.4
2. B.5
3. C.6
4. D.3

Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k},\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\vec{a}\times \vec{b}$. Let $\vec{d}$ be a vector such that $|\vec{d}-\vec{a}|=\sqrt{11},|\vec{c}\times \vec{d}|=3$ and the angle between $\vec{c}$ and $\vec{d}$ is $\frac{\pi }{4}$. Then $\vec{a}\cdot \vec{d}$ is equal to

1. A.0
2. B.1
3. C.3
4. D.11

Let the lines ${\mathrm{L}}_1:\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda (2\hat{i}+3\hat{j}+4\hat{k}),\lambda \in \mathbb{R}$ and ${\mathrm{L}}_2:\vec{r}=(4\hat{i}+\hat{j})+\mu (5\hat{i}+2\hat{j}+\hat{k}),\mu \in \mathbb{R}$, intersect at the point R . Let P and Q be the points lying on lines $L_1$ and $L_2$, respectively, such that $|\overrightarrow{\mathrm{PR}}|=\sqrt{29}$ and $|\overrightarrow{\mathrm{PQ}}|=\sqrt{\frac{47}{3}}$. If the point P lies in the first octant, then $27(\mathrm{QR}{)}^2$ is equal to

1. A.360
2. B.348
3. C.320
4. D.340

Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\vec{a}\times \vec{b}=2(\vec{a}\times \vec{c})$. If $|\vec{a}|=1,|\vec{b}|=4,|\vec{c}|=2$, and the angle between $\vec{b}$ and $\vec{c}$ is $60^{\circ }$, then $|\vec{a}\cdot \vec{c}|$ is equal to

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

Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$, $\vec{c}=\lambda \hat{i}+\hat{j}+\hat{k}$ and $\vec{v}=\vec{a}\times \vec{b}$. If $\vec{v}\cdot \vec{c}=11$ and the length of the projection of $\vec{b}$ on $\vec{c}$ is $p$, then $9p^2$ is equal to

1. A.9
2. B.6
3. C.4
4. D.12

Let $\overrightarrow{\mathrm{AB}}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{i}+2\hat{j}+\lambda \hat{k},\lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v}=\hat{i}+\hat{j}+\hat{k}$ on the diagonal $\overrightarrow{\mathrm{AC}}$ of the parallelogram ABCD be of length one unit. If $\alpha ,\beta$, where $\alpha >\beta$, be the roots of the equation ${\lambda }^2{x}^2-6\lambda x+5=0$, then $2\alpha -\beta$ is equal to

1. A.3
2. B.6
3. C.4
4. D.1

Let $\vec{\mathrm{a}}=-\hat{i}+2\hat{j}+2\hat{k}$, $\vec{\mathrm{b}}=8\hat{i}+7\hat{j}-3\hat{k}$ and $\vec{c}$ be a vector such that $\vec{\mathrm{a}}\times \vec{c}=\vec{\mathrm{b}}$. If $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=4$, then $|\vec{\mathrm{a}}+\vec{c}{|}^2$ is equal to :

1. A.30
2. B.33
3. C.27
4. D.35

Let $(\alpha ,\beta ,\gamma )$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on the line $\vec{\mathrm{r}}=(-\hat{i}+3\hat{j}+\hat{k})+\lambda (2\hat{i}+3\hat{j}-\hat{k})$. Then the length of the projection of the vector $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ on the vector $6\hat{i}+2\hat{j}+3\hat{k}$ is :

1. A.$\frac{18}{7}$
2. B.$\frac{15}{7}$
3. C.4
4. D.3

Let $\vec{c}$ and $\vec{d}$ be vectors such that $|\vec{c}+\vec{d}|=\sqrt{29}$ and $\vec{c}\times (2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times \vec{d}$. If ${\lambda }_1,{\lambda }_2({\lambda }_1>{\lambda }_2)$ are the possible values of $(\vec{c}+\vec{d})\cdot (-7\hat{i}+2\hat{j}+3\hat{k})$, then the equation ${\mathrm{K}}^2{x}^2+({\mathrm{K}}^2-5\mathrm{K}+{\lambda }_1)xy+\left(3\mathrm{K}+\frac{{\lambda }_2}{2}\right)y^2-8x+12y+{\lambda }_2=0$ represents a circle, for K equal to :

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

Let the line $L_1$ be parallel to the vector $-3\hat{i}+2\hat{j}+4\hat{k}$ and pass through the point $(2,6,7)$, and the line $L_2$ be parallel to the vector $2\hat{i}+\hat{j}+3\hat{k}$ and pass through the point $(4,3,5)$. If the line $L_3$ is parallel to the vector $-3\hat{i}+5\hat{j}+16\hat{k}$ and intersects the lines $L_1$ and $L_2$ at the points $C$ and $D$, respectively, then ${\left|\overrightarrow{CD}\right|}^2$ is equal to:

1. A.290
2. B.171
3. C.89
4. D.312

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is:

1. A.$\frac{1}{\sqrt{2}}(-\hat{i}+\hat{k})$
2. B.$\frac{1}{\sqrt{5}}(\hat{j}-2\hat{k})$
3. C.$\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$
4. D.$\frac{1}{\sqrt{3}}(-\hat{i}+\hat{j}-\hat{k})$

Consider two vectors $\vec{u}=3\hat{i}-\hat{j}$ and $\vec{v}=2\hat{i}+\hat{j}-\lambda \hat{k},\lambda >0$. The angle between them is given by ${\cos }^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v}={\vec{v}}_1+{\vec{v}}_2$, where ${\vec{v}}_1$ is parallel to $\vec{u}$ and ${\vec{v}}_2$ is perpendicular to $\vec{u}$. Then the value of ${\left|{\vec{v}}_1\right|}^2+{\left|{\vec{v}}_2\right|}^2$ is equal to

1. A.$\frac{23}{2}$
2. B.$\frac{25}{2}$
3. C.$10$
4. D.$14$

Let the three sides of a triangle ABC be given by the vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let $G$ be the centroid of the triangle $ABC$. Then $6\left(|\overrightarrow{\mathrm{AG}}{|}^2+|\overrightarrow{\mathrm{BG}}{|}^2+|\overrightarrow{\mathrm{CG}}{|}^2\right)$ is equal to ____ .

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=3\hat{i}+2\hat{j}-\hat{k},\vec{c}=\lambda \hat{j}+\mu \hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a}\times \hat{d}=\vec{b}\times \hat{d}$ and $\vec{c}\cdot \hat{d}=1$. If $\vec{c}$ is perpendicular to $\vec{a}$, then $|3\lambda \hat{d}+\mu \vec{c}{|}^2$ is equal to____