JEE MAIN Vector Previous Year Questions (PYQs) – Page 7 of 17

Let $\vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}$, $\vec{b} = 3\hat{i} + 5\hat{k}$ and $\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c} \cdot \vec{d} = 12$. Then $(-\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d})$ is equal to:

Let $P(3,2,3)$, $Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\triangle PQR$. Then, the angle $\angle QPR$ is:

Let $\vec a=6\hat i+\hat j-\hat k$ and $\vec b=\hat i+\hat j$. If $\vec c$ is a vector such that $|\vec c|\ge 6$, $\ \vec a\cdot\vec c=6|\vec c|$, $|\vec c-\vec a|=2\sqrt2$ and the angle between $\vec a\times\vec b$ and $\vec c$ is $60^\circ$, then $|(,(\vec a\times\vec b)\times\vec c,)|$ equals:

Let $\vec{a} = \hat{i} + \hat{j} + \hat{k}$, $\vec{c} = \hat{j} - \hat{k}$, and a vector $\vec{b}$ be such that $\vec{a} \times \vec{b} = \vec{c}$ and $\vec{a} \cdot \vec{b} = 3$. Then $|\vec{b}|$ equals:

Let the position vectors of two points P and Q be 3$\widehat i$ $-$ $\widehat j$ + 2$\widehat k$ and $\widehat i$ + 2$\widehat j$ $-$ 4$\widehat k$, 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 :

If the two lines ${l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2$ and ${l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over 2}$ are perpendicular, then an angle between the lines l₂ and ${l_3}:{{1 - x} \over 3} = {{2y - 1} \over { - 4}} = {z \over 4}$ is :

Let $\vec{a},\vec{b},\vec{c}$ be three coplanar concurrent vectors such that the angles between any two of them are the same. If the product of their magnitudes is $14$ and $ (\vec{a}\times\vec{b})\cdot(\vec{b}\times\vec{c}) +(\vec{b}\times\vec{c})\cdot(\vec{c}\times\vec{a}) +(\vec{c}\times\vec{a})\cdot(\vec{a}\times\vec{b})=168, $ then $|\vec{a}|+|\vec{b}|+|\vec{c}|$ is equal to:

If the points $\mathbf{P}$ and $\mathbf{Q}$ are respectively the circumcenter and the orthocentre of a $\triangle ABC$, then $\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC}$ is equal to:

>Let $A, B, C$ be three points in xy-plane, whose position vector are given by $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $a \hat{i}+(1-a) \hat{j}$ respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$ is $\frac{9}{\sqrt{2}}$, then the sum of all the possible values of $a$ is :

Let $\overrightarrow{OA}=\vec a,\ \overrightarrow{OB}=12\vec a+4\vec b$ and $\overrightarrow{OC}=\vec b$, where $O$ is the origin. If $S$ is the parallelogram with adjacent sides $OA$ and $OC$, then $\dfrac{\text{area of the quadrilateral }OABC}{\text{area of }S}$ is equal to: