JEE MAIN Vector Previous Year Questions (PYQs) – Page 5 of 17

Let $\vec a = 3\hat{i}-\hat{j}+2\hat{k}$, $\vec b=\vec a \times (\hat{i}-2\hat{k})$ and $\vec c=\vec b \times \hat{k}$. Then the projection of $\vec c-2\hat{j}$ on $\vec a$ is:

$ \text{If the points with position vectors } \alpha\hat{i}+10\hat{j}+13\hat{k},; 6\hat{i}+11\hat{j}+11\hat{k},; \dfrac{9}{2}\hat{i}+\beta\hat{j}-8\hat{k} \text{ are collinear, then } (19\alpha-6\beta)^2 \text{ is equal to:} $

Let $\sqrt{3}\,\hat{i}+\hat{j}$, $\ \hat{i}+\sqrt{3}\,\hat{j}$ and $\ \beta\,\hat{i}+(1-\beta)\,\hat{j}$ respectively be the position vectors of the points $A$, $B$ and $C$ with respect to the origin $O$. If the distance of $C$ from the bisector of the acute angle between $\overrightarrow{OA}$ and $\overrightarrow{OB}$ is $\dfrac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is:

$ \text{Let } S \text{ be the set of all } a \in \mathbb{R} \text{ for which the angle between the vectors } \vec{u}=a(\log_{e} b),\hat{i}-6\hat{j}+3\hat{k} \text{ and } \vec{v}=(\log_{e} b),\hat{i}+2\hat{j}+2a(\log_{e} b),\hat{k},\ (b>1), \text{ is acute. Then } S \text{ is equal to:} $

The position vectors of the vertices $A,B,C$ of a triangle are $2\hat i-3\hat j+3\hat k$, $2\hat i+2\hat j+3\hat k$ and $-\hat i+\hat j+3\hat k$ respectively. Let $l$ denote the length of the angle bisector $AD$ of $\angle BAC$ (where $D$ is on the line segment $BC$). Then $2l^{2}$ equals:

et $A(x,y,z)$ be a point in $xy$-plane, which is equidistant from three points $(0,3,2)$, $(2,0,3)$ and $(0,0,1)$. Let $B=(1,4,-1)$ and $C=(2,0,-2)$. Then among the statements (S1): $\triangle ABC$ is an isosceles right angled triangle, and (S2): the area of $\triangle ABC$ is $\dfrac{9\sqrt{2}}{2}$,

Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$. If $\vec{u}$ is perpendicular to $\vec{a}$ and $\vec{u}\cdot\vec{b}=24$, then $\lvert\vec{u}\rvert^{2}$ is equal to :

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

Let $\vec a,\vec b,\vec c$ be three non-zero vectors such that $\vec b$ and $\vec c$ are non-collinear. If $\vec a+5\vec b$ is collinear with $\vec c$, $\ \vec b+6\vec c$ is collinear with $\vec a$ and $\vec a+\alpha\vec b+\beta\vec c=\vec 0$, then $\alpha+\beta$ is equal to: