JEE Main 2019 — Vectors & 3D Geometry

Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point $R$. The reflection of $R$ in the $x y$ - plane has coordinates:

The direction ratios of normal to the plane through the points (0,-1,0) and (0,0,1) and making an angle $\frac{\pi}{4}$ with the plane $y-z+5=0$ are; 2,-1,1 $2, \sqrt{2}-\sqrt{2}$ $\sqrt{2}, 1,-1$ $2 \sqrt{3}, 1,-1$

Let $\vec{a}=\hat{i}+\hat{j}+\sqrt{2}\hat{k},$ $\vec{b}={b}_{1}\hat{i}+{b}_{2}\hat{j}+\sqrt{2}\hat{k}$ and $\vec{c}=5\hat{i}+\hat{j}+\sqrt{2}\hat{k}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $|\vec{a}|$ . If $\vec{a}+\vec{b}$ is perpendicular to $\vec{c}$ , then $|\vec{b}|$ is equal to:

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\vec{b}$ and $\vec{c}$ respectively and $\vec{a}\times (\vec{b}\times \vec{c})=\frac{1}{2}\vec{b}$, then $|\alpha -\beta |$ is equal to :

Let $\vec{a}=2\hat{i}+{\lambda }_{1}\hat{j}+3\hat{k}, \vec{b}=4\hat{i}+(3-{\lambda }_{2})\hat{j}+6\hat{k}$ and $\vec{c}=3\hat{i}+6\hat{j}+({\lambda }_{3}-1)\hat{k}$ be three vectors such that $\vec{b}=2\vec{a}$ and $\vec{a}$ is perpendicular to $\vec{c}.$ Then a possible value of $({\lambda }_{1},{ \lambda }_{2}, {\lambda }_{3})$ is

Let $A(3,0,-1), B(2,10,6)$ and $C(1,2,1)$ be the vertices of a triangle and $M$ be the mid-point of $AC$. If $G$ divides $BM$ in the ratio, $2:1$ , then $cos(\angle GOA)$ ($O$ being the origin) is equal to

Let $\vec{a}=\hat{i}+2 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}+\lambda \hat{j}+4 \hat{k}$ and $\vec{c}=2 \hat{i}+4 \hat{j}+\left(\lambda^{2}-1\right) \hat{k}$ be coplanar vectors. Then the non-zero vector $\vec{a} \times \vec{c}$ is:

If the lines $x=ay+b,z=cy+d$ and $x={a}^{'}z+{b}^{'}, y={c}^{'} z+{d}^{'}$ are perpendicular, then

Let $\vec{a}=\hat{i}-\hat{j}, \vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times \vec{c}+\vec{b}=\vec{0}$ and $\vec{a}.\vec{c}=4,$ then ${|\vec{c}|}^{2}$ is equal to:

Let $\overset{\rightarrow }{\alpha }=3\hat{i}+\hat{j}$ and $\vec{\beta }=2\hat{i}-\hat{j}+3\hat{k}.$ If $\vec{\beta }=\vec{{\beta }_{1}}-\vec{{\beta }_{2}},$ where $\vec{{\beta }_{1}}$ is parallel to $\overset{\rightarrow }{\alpha }$ and $\vec{{\beta }_{2}}$ is perpendicular to $\vec{\alpha },$ then $\vec{{\beta }_{1}}\times \vec{{\beta }_{2}}$ is equal to:

The length of the perpendicular from the point $(2,-1, 4)$ on the straight line $\frac{x+3 }{10}=\frac{y-2}{-7}=\frac{z}{1}$ is

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:

The distance of the point having position vector $-\hat{i}+2\hat{j}+6\hat{k}$ from the straight line passing through the point $(2, 3, -4)$ and parallel to the vector, $6\hat{i}+3\hat{j}-4\hat{k}$ is

Let $\vec{a}=3\hat{i}+2\hat{j}+x\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k},$ for some real $x$. Then the condition for $|\vec{a}\times \vec{b} |=r$to follow

The equation of the line passing through $(-4, 3, 1),$ parallel to the plane $x+2y-z-5=0$ and intersecting the line $\frac{x + 1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is

If the length of the perpendicular from the point $(\beta ,0,\beta ),(\beta \neq 0)$ to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is $\sqrt{\frac{3}{2}}$ , then $\beta$ is equal to

Let $\alpha \in R$ 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 = {$\alpha :\vec{a},\vec{b}$ and $\vec{c}$ are coplanar}

If a unit vector $\vec{a}$ makes angles $\frac{\pi }{3}$ with $\hat{i},\frac{\pi }{4}$ with $\hat{j}$ and $\theta \in (0,\pi )$ with $\hat{k},$ then a value of $\theta$ is:

The magnitude of the projection of the vector $2\hat{i}+3\hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k},$ is:

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 $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ with respect to the origin $\mathrm{O}$. If the distance of $\mathrm{C}$ from the bisector of the acute angle between $\mathrm{OA}$ and $\mathrm{OB}$ is $\frac{3}{\sqrt{2}},$ then the sum of all possible values of $\beta$ is:

The vertices $B$ and $C$ of a $\Delta ABC$ lie on the line, $\frac{x+2}{3}=\frac{y-1}{0}=\frac{z}{4}$ such that $BC=5$ units. Then the area (in sq. units) of this triangle, given the point $A(1, -1, 2),$ is

The sum of the distinct real values of $\mu$ for which the vectors $\mu \hat{i}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}$ are co-planar, is

Let $A$ be a point on the line $\vec{r}=(1-3\mu )\hat{i}+(\mu -1)\hat{j}+(2+5\mu )\hat{k}$ and $B(3, 2, 6)$ be a point in the space. Then the value of $\mu$ for which the vector $\vec{AB}$ is parallel to the plane $x-4y+3z=1$ is

If a point $R(4,y,z)$ lies on the line segment joining the points $P(2,-3,4)$ and $Q(8,0,10),$ then the distance of $R$ from the origin is

Let $\vec{\alpha }=(\lambda -2) \vec{a}+\vec{b}$ and $\vec{\beta }=(4\lambda -2) \vec{a}+3\vec{b},$ be two given vectors where vectors $\vec{a}$ and $\vec{b}$ are non-collinear. The value of $\lambda$ for which vectors $\vec{\alpha }$ and $\vec{\beta }$ are collinear, is: