Vector Algebra · JEE Main previous year questions

If the vectors $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}+\hat{k}$ and $\vec{c}=\lambda \hat{i}+\hat{j}+\mu \hat{k}$ are mutually orthogonal, then $(\lambda ,\mu )$ is equal to :

1. A.$(2,-3)$
2. B.$(-2,3)$
3. C.$(3,-2)$
4. D.$(-3,2)$

Let $\vec{a}=\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$. Then the vector $\vec{b}$ satisfying $\vec{a}\times \vec{b}+\vec{c}=\vec{0}$ and $\vec{a}\cdot \vec{b}=3$ :

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

The projections of a vector on the three coordinate axis are $6,-3,2$ respectively. The direction cosines of the vector are :

1. A.$\frac{6}{5},\frac{-3}{5},\frac{2}{5}$
2. B.$\frac{6}{7},\frac{-3}{7},\frac{2}{7}$
3. C.$\frac{-6}{7},\frac{-3}{7},\frac{2}{7}$
4. D.$6,-3,2$

If $\vec{u},\vec{v},\vec{w}$ are non-coplanar vectors and $p,q$ are real numbers, then the equality $$\left[3\vec{u}p\vec{v}p\vec{w}\right]-\left[p\vec{v}\vec{w}q\vec{u}\right]-\left[2\vec{w}q\vec{v}q\vec{u}\right]=0$$ holds for :

1. A.exactly two values of $(p,q)$
2. B.more than two but not all values of $(p,q)$
3. C.all values of $(p,q)$
4. D.exactly one value of $(p,q)$

The values of a, for which the points $A,B,C$ with position vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of a right angled triangle with $C=\frac{\pi }{2}$ are :

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

If $(\vec{a}\times \vec{b})\times \vec{c}=\vec{a}\times (\vec{b}\times \vec{c})$ where $\vec{a},\vec{b}$ and $\vec{c}$ are any three vectors such that $\vec{a}\cdot \vec{b}\ne 0$, $\vec{b}\cdot \vec{c}\ne 0$ then $\vec{a}$ and $\vec{c}$ are :

1. A.inclined at an angle of $\frac{\pi }{3}$ between them
2. B.inclined at an angle of $\frac{\pi }{6}$ between them
3. C.perpendicular
4. D.parallel

For any vector $\vec{a}$, the value of ${\left(\vec{a}\times \hat{i}\right)}^2+{\left(\vec{a}\times \hat{j}\right)}^2+{\left(\vec{a}\times \hat{k}\right)}^2$ is equal to :

1. A.$3{\vec{a}}^2$
2. B.${\vec{a}}^2$
3. C.$2{\vec{a}}^2$
4. D.$4{\vec{a}}^2$

If $\vec{a},\vec{b},\vec{c}$ are non coplanar vectors and $\lambda$ is a real number then $$\left[\lambda \left(\vec{a}+\vec{b}\right){\lambda }^2\vec{b}\lambda \vec{c}\right]=\left[\vec{a}\vec{b}+\vec{c}\vec{b}\right]\text{ for :}$$

1. A.exactly one value of $\lambda$
2. B.no value of $\lambda$
3. C.exactly three values of $\lambda$
4. D.exactly two values of $\lambda$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of these are collinear. If the vector $\vec{a}+2\vec{b}$ is collinear with $\vec{c}$ and $\vec{b}+3\vec{c}$ is collinear with $\vec{a}$ ($\lambda$ being some non-zero scalar) then $\vec{a}+2\vec{b}+6\vec{c}$ equals to :

1. A.$\vec{0}$
2. B.$\lambda \vec{b}$
3. C.$\lambda \vec{c}$
4. D.$\lambda \vec{a}$

Let $\vec{u},\vec{v},\vec{w}$ be such that $|\vec{u}|=1,|\vec{v}|=2,|\vec{w}|=3$. If the projection of $\vec{v}$ along $\vec{u}$ is equal to that of $\vec{w}$ along $\vec{u}$ and $\vec{v},\vec{w}$ are perpendicular to each other then $|\vec{u}-\vec{v}+\vec{w}|$ equals :

1. A.$14$
2. B.$\sqrt{7}$
3. C.$\sqrt{14}$
4. D.$2$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be non-zero vectors such that $(\vec{a}\times \vec{b})\times \vec{c}=\frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$. If $\theta$ is the acute angle between the vectors $\vec{b}$ and $\vec{c}$, then $\sin \theta$ equals :

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

If $\vec{a},\vec{b},\vec{c}$ are non-coplanar vectors and $\lambda$ is a real number, then the vectors $\vec{a}+2\vec{b}+3\vec{c},\lambda \vec{b}+4\vec{c}$ and $(2\lambda -1)\vec{c}$ are non coplanar for :

1. A.no value of $\lambda$
2. B.all except one value of $\lambda$
3. C.all except two values of $\lambda$
4. D.all values of $\lambda$

$I_n={\int }_0^{\pi /4}{\tan }^{n}xdx$ then ${\lim }_{n\to \infty }n[I_n+I_{n+2}]$ equals

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

If $|\vec{a}|=4,|\vec{b}|=2$ and the angle between $\vec{a}$ and $\vec{b}$ is $\pi /6$ then $(\vec{a}\times \vec{b}{)}^2$ is equal to :

1. A.$48$
2. B.$16$
3. C.$\vec{a}$
4. D.none of these

If the vectors $\vec{\mathbf{a}},\vec{\mathbf{b}}$ and $\vec{\mathbf{c}}$ form the sides $BC,CA$ and $AB$ respectively of a triangle $ABC$, then :

1. A.$\vec{\mathbf{a}}\cdot \vec{\mathbf{b}}+\vec{\mathbf{b}}\cdot \vec{\mathbf{c}}+\vec{\mathbf{c}}\cdot \vec{\mathbf{a}}=0$
2. B.$\vec{\mathbf{a}}\times \vec{\mathbf{b}}=\vec{\mathbf{b}}\times \vec{\mathbf{c}}=\vec{\mathbf{c}}\times \vec{\mathbf{a}}$
3. C.$\vec{\mathbf{a}}\cdot \vec{\mathbf{b}}=\vec{\mathbf{b}}\cdot \vec{\mathbf{c}}=\vec{\mathbf{c}}\cdot \vec{\mathbf{a}}=0$
4. D.$\vec{\mathbf{a}}\times \vec{\mathbf{b}}+\vec{\mathbf{b}}\times \vec{\mathbf{c}}+\vec{\mathbf{c}}\times \vec{\mathbf{a}}=\vec{0}$

If the vectors $\vec{c},\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$ and $\hat{b}=\hat{j}$ are such that $\vec{a},\vec{c}$ and $\vec{b}$ form a right handed system then $\vec{c}$ is :

1. A.$z\hat{i}-x\hat{k}$
2. B.$\vec{0}$
3. C.$y\hat{j}$
4. D.$-z\hat{i}+x\hat{k}$

$\vec{a}=3\hat{i}-5\hat{j}$ and $\vec{b}=6\hat{i}+3\hat{j}$ are two vectors and $\vec{c}$ is a vector such that $\vec{c}=\vec{a}\times \vec{b}$ then $|\vec{a}|:|\vec{b}|:|\vec{c}|=$

1. A.$\sqrt{34}:\sqrt{45}:\sqrt{39}$
2. B.$\sqrt{34}:\sqrt{45}:39$
3. C.$34:39:45$
4. D.$39:35:34$

If $\vec{a},\vec{b},\vec{c}$ are vectors such that $[\vec{a}\vec{b}\vec{c}]=4$ then $[\vec{a}\times \vec{b}\vec{b}\times \vec{c}\vec{c}\times \vec{a}]=$

1. A.$16$
2. B.$64$
3. C.$4$
4. D.$8$