If the middle points of sides BC, CA \ and AB of triangle ABC are respectively D, E, F then position vector of centre of triangle DEF , when position vector of A, B, C are respectively i + j , j + k , k + i is

Correct answer: D. (2)/(3)( i + j + k ) The position vector of points D, E, F are respectively i + j + k , i + k + j 2 and i + k 2 + j So, position vector of centre of Δ DEF=(1)/(3)[ i + j 2 + k + i k + j 2 + i + k 2 + j ]=(2)/(3)[ i + j + k ]

BITSAT 2014

Exam: BITSAT
Year: 2014
Subject: Mathematics
Chapter: Vector Algebra

BITSAT 2014 · Mathematics · Vector Algebra

If the middle points of sides BC, CA \ and AB of triangle ABC are...

If the middle points of sides

\mathrm{BC}, \mathrm{CA} \ \text{ and } \mathrm{AB}

of triangle

\mathrm{ABC}

are respectively

\mathrm{D}, \mathrm{E}, \mathrm{F}

then position vector of centre of triangle

\mathrm{DEF}

, when position vector of

A, \quad B, \quad C

are respectively

\hat{i}+\hat{j}, \hat{j}+\hat{k}, \hat{k}+\hat{i}

is $If the middle points of sides BC, CA \ and AB of triangle ABC are respectively D, E, F then position vector of centre$

Options

A $\frac{1}{3}(\hat{i}+\hat{j}+\hat{k})$
B $(\hat{i}+\hat{j}+\hat{k})$
C $2(\hat{i}+\hat{j}+\hat{k})$
D $\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})$

Correct answer

D. $\frac{2}{3}(\hat{i}+\hat{j}+\hat{k})$

Step-by-step solution

The position vector of points

D, E, F

are respectively

\hat{i}+\hat{j}+\hat{k}, \hat{i}+\frac{\hat{k}+\hat{j}}{2}

and

\frac{\hat{i}+\hat{k}}{2}+\hat{j}

$If the middle points of sides BC, CA \ and AB of triangle ABC are respectively D, E, F then position vector of centre$
So, position vector of centre of

\Delta \mathrm{DEF}=\frac{1}{3}\left[\frac{\hat{i}+\hat{j}}{2}+\hat{k}+\hat{i} \frac{\hat{k}+\hat{j}}{2}+\frac{\hat{i}+\hat{k}}{2}+\hat{j}\right]=\frac{2}{3}[\hat{i}+\hat{j}+\hat{k}]

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