$OABC$ is a regular tetrahedron of side unity, then

Important Questions on Three Dimensional Coordinate System

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 2>Q 121

If $OABC$ is a tetrahedron such that ${\left|\overrightarrow{OA}\right|}^2+{\left|\overrightarrow{BC}\right|}^2={\left|\overrightarrow{OB}\right|}^2+{\left|\overrightarrow{CA}\right|}^2={\left|\overrightarrow{OC}\right|}^2+{\left|\overrightarrow{AB}\right|}^2$, then

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 2>Q 122

If the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ intersects the line $3{\beta }^{2}x+3\left(1-2\alpha \right)y+z=3=-\frac{1}{2}\left\{6{\alpha }^{2}x+3\left(1-2\beta \right)y+2z\right\}$ then point $\left(\alpha ,\beta ,1\right)$ lie on the plane

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 2>Q 123

Let $PM$ be the perpendicular from the point $P(1,2,3)$ to $XY$ plane. If $\overrightarrow{OP}$ makes an angle $\theta$ with the positive direction of $Z$-axis and $\overrightarrow{OM}$ makes an angle $\varphi$ with the positive direction of $X$-axis, where $O$ is the origin and $\theta$ and $\varphi$ are acute angles, then

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 2>Q 124

A variable plane which remains at a constant distance $p$ from the origin $\left(0\right)$ cuts the coordinate axes in $A,B,C$

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 7>Q 208

Find the angle between the lines whose direction cosines has the relation $l+m+n=0$ and $2l^2+2m^2-n^2=0$

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 7>Q 209

Prove that the two lines whose direction cosines are connected by the two relations $al+bm+cn=0$ and $u{l}^2+v{m}^2+w{n}^2=0$ are perpendicular if $a^2\left(v+w\right)+b^2\left(w+u\right)+c^2\left(u+v\right)=0$ and parallel if $\frac{a^2}{u}+\frac{b^2}{v}+\frac{c^2}{w}=0$

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 7>Q 210

Find the point on the line $\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}$ at a distance of $3\sqrt{2}$ from the point $\left(1,2,3\right)$

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 3 - Three Dimensional Coordinate System>Three Dimensional Coordinate System Exercise 7>Q 211

A line passes through $\left(2,-1,3\right)$ and is perpendicular to the lines $\overrightarrow{r}=\left(\hat{i}+\hat{j}-\hat{k}\right)+\lambda \left(2\hat{i}-2\hat{j}+\hat{k}\right)$ and $\overrightarrow{r}=\left(2\hat{i}-\hat{j}-3\hat{k}\right)+\mu \left(\hat{i}+2\hat{j}+2\hat{k}\right)$ obtain its equation.