Let the position vectors of the vertices A, B | vedclass

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

Question

(B) The vertices are $A(2, 2, 1)$,$B(1, 2, 2)$,and $C(2, 1, 2)$.Calculate the side lengths:$AB = \sqrt{(1-2)^2 + (2-2)^2 + (2-1)^2} = \sqrt{1+0+1} = \

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(B) The vertices are $A(2, 2, 1)$,$B(1, 2, 2)$,and $C(2, 1, 2)$.Calculate the side lengths:$AB = \sqrt{(1-2)^2 + (2-2)^2 + (2-1)^2} = \sqrt{1+0+1} = \sqrt{2}$$BC = \sqrt{(2-1)^2 + (1-2)^2 + (2-2)^2} = \sqrt{1+1+0} = \sqrt{2}$$CA = \sqrt{(2-2)^2 + (2-1)^2 + (1-2)^2} = \sqrt{0+1+1} = \sqrt{2}$Since $AB = BC = CA = \sqrt{2}$,the triangle is equilateral.For an equilateral triangle,the orthocenter $H$ coincides with the centroid $G$.$G = \left(\frac{2+1+2}{3}, \frac{2+2+1}{3}, \frac{1+2+2}{3}ight) = \left(\frac{5}{3}, \frac{5}{3}, \frac{5}{3}ight)$.In an equilateral triangle,the length of the perpendicular from the centroid to any side is the distance from the centroid to the midpoint of that side.For side $AB$,the midpoint $D$ is $\left(\frac{2+1}{2}, \frac{2+2}{2}, \frac{1+2}{2}ight) = \left(\frac{3}{2}, 2, \frac{3}{2}ight)$.$l_1 = 	ext{distance } GD = \sqrt{\left(\frac{5}{3}-\frac{3}{2}ight)^2 + \left(\frac{5}{3}-2ight)^2 + \left(\frac{5}{3}-\frac{3}{2}ight)^2}$$l_1 = \sqrt{\left(\frac{1}{6}ight)^2 + \left(-\frac{1}{3}ight)^2 + \left(\frac{1}{6}ight)^2} = \sqrt{\frac{1}{36} + \frac{4}{36} + \frac{1}{36}} = \sqrt{\frac{6}{36}} = \sqrt{\frac{1}{6}}$.Since the triangle is equilateral,$l_1 = l_2 = l_3 = \sqrt{\frac{1}{6}}$.Therefore,$l_1^2 + l_2^2 + l_3^2 = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$.

Similar Questions

If $\alpha, \beta, \gamma$ are real numbers such that $(\frac{7}{3}+\beta) \hat{i}-\hat{j}+(\alpha+\gamma) \hat{k}=\frac{5}{3}(\alpha \hat{i}+\hat{j}-\hat{k})+\beta(2 \hat{j}+\hat{k})+(\hat{i}+\gamma \hat{j}+3 \hat{k})$, then $5 \alpha-9 \beta+13 \gamma=$

If the sum of two unit vectors is a unit vector,then the magnitude of their difference is

If $\vec{a} + 5\vec{b} = \vec{c}$ and $\vec{a} - 7\vec{b} = 2\vec{c}$,then which of the following is true?

Let $ABCDEF$ be a regular hexagon with the vertices $A, B, C, D, E$ and $F$ in counter-clockwise order. Then,the vector $\vec{AB} + \vec{BC}$ is parallel to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module