O(0,0,0), A(3,1,4), B(1,3,2) and C(0,4,-2) ar | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The vertices of the tetrahedron are $O(0,0,0), A(3,1,4), B(1,3,2)$, and $C(0,4,-2)$.The centroid $G$ of the tetrahedron is given by the average of

Vedclass · Accepted Answer

$\left(\frac{10}{9}, \frac{20}{9}, \frac{10}{9}\right)$

Vedclass · Answer

(C) The vertices of the tetrahedron are $O(0,0,0), A(3,1,4), B(1,3,2)$, and $C(0,4,-2)$.The centroid $G$ of the tetrahedron is given by the average of its vertices: $G = \left(\frac{0+3+1+0}{4}, \frac{0+1+3+4}{4}, \frac{0+4+2-2}{4}\right) = \left(1, 2, 1\right)$.The centroid $G_1$ of the face $ABC$ is given by the average of vertices $A, B$, and $C$: $G_1 = \left(\frac{3+1+0}{3}, \frac{1+3+4}{3}, \frac{4+2-2}{3}\right) = \left(\frac{4}{3}, \frac{8}{3}, \frac{4}{3}\right)$.We need to find the point $P$ that divides the line segment $GG_1$ in the ratio $1:2$. Using the section formula, $P = \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n}\right)$ with $m=1, n=2$, $G(1,2,1)$ and $G_1(\frac{4}{3}, \frac{8}{3}, \frac{4}{3})$:$x = \frac{1(\frac{4}{3}) + 2(1)}{1+2} = \frac{\frac{4}{3} + 2}{3} = \frac{10}{9}$.$y = \frac{1(\frac{8}{3}) + 2(2)}{1+2} = \frac{\frac{8}{3} + 4}{3} = \frac{20}{9}$.$z = \frac{1(\frac{4}{3}) + 2(1)}{1+2} = \frac{\frac{4}{3} + 2}{3} = \frac{10}{9}$.Thus, the point is $\left(\frac{10}{9}, \frac{20}{9}, \frac{10}{9}\right)$.

$O(0,0,0), A(3,1,4), B(1,3,2)$ and $C(0,4,-2)$ are the vertices of a tetrahedron. If $G$ is the centroid of the tetrahedron and $G_1$ is the centroid of its face $ABC$, then the point which divides $GG_1$ in the ratio $1:2$ is

Similar Questions

The harmonic conjugate of $P(-9, 12, -15)$ with respect to the line segment $AB$,where $A=(1, -2, 3)$ and $B=(-4, 5, -6)$ is

The $x-$coordinate of a point on the line joining the points $Q(2, 2, 1)$ and $R(5, 1, -2)$ is $4$. Find its $z-$coordinate.

Let $P(\alpha, 4, 7)$ and $Q(3, \beta, 8)$ be two points. If the $YZ$-plane divides the line segment joining $P$ and $Q$ in the ratio $2:3$ and the $ZX$-plane divides the line segment joining $P$ and $Q$ in the ratio $4:5$,then the length of the line segment $PQ$ is:

The point dividing the join of $(3, -2, 1)$ and $(-2, 3, 11)$ in the ratio $2:3$ is

If $A (1, 2, -1)$ and $B (-1, 0, 1)$ are two points,then the coordinates of the point which divides $AB$ externally in the ratio $1 : 2$ are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module