+91-8076753736[email protected]
logo
logo
-->

Vector Algebra

Question
CBSEENMA12034280
Wired Faculty App

If D, E, F are the mid points of the sides of triangle ABC, prove that
area left parenthesis increment DEF right parenthesis space equals space 1 fourth area space left parenthesis increment thin space ABC right parenthesis.

Solution
Take A as origin. Let position vectors of B and C be straight b with rightwards arrow on top space and space straight c with rightwards arrow on top respectively.
therefore  position vectors of D, E, F are 1 half left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis comma space space space space 1 half straight c with rightwards arrow on top space and space space 1 half straight b with rightwards arrow on top space respectively.
therefore space space DE with rightwards arrow on top space equals space left parenthesis straight P. straight V. space of space straight E right parenthesis space minus space left parenthesis straight P. straight V. space of space straight D right parenthesis space space space space space space space space space space space space space equals space 1 half straight c with rightwards arrow on top space minus space 1 half left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space equals space minus 1 half straight b with rightwards arrow on top and space DF with rightwards arrow on top space equals space left parenthesis straight P. straight V. space of space straight F right parenthesis space space minus space left parenthesis straight P. straight V. space of space straight D right parenthesis space space space space space space space space space space space space space space equals space 1 half straight b with rightwards arrow on top space minus space 1 half left parenthesis straight b with rightwards arrow on top plus straight c with rightwards arrow on top right parenthesis space equals space minus 1 half straight c with rightwards arrow on top

therefore space space vector space area space of space increment DEF space equals space 1 half DE with rightwards arrow on top space cross times space DF with rightwards arrow on top space space equals space 1 half open parentheses negative 1 half straight b with rightwards arrow on top close parentheses space cross times space open parentheses negative 1 half straight c with rightwards arrow on top close parentheses space equals 1 fourth space open parentheses 1 half space left parenthesis straight b with rightwards arrow on top space cross times space stack straight c right parenthesis with rightwards arrow on top close parentheses space equals space 1 fourth left parenthesis vector space area space of space increment ABC right parenthesis.
Hence the result. 

Some More Questions From Vector Algebra Chapter

Classify the following measures as scalars and vectors.
(i) 5 seconds
(ii) 1000 cm3
(iii) 10 Newton
(iv) 30 km hr
(v) 10 g/cm3
(vi) 20 m/s towards north

Classify the following measures as scalars and vectors.
10 kg

Classify the following measures as scalars and vectors.
2 meters north-west

Classify the following measures as scalars and vectors.
40°

Classify the following measures as scalars and vectors.
40 watt

Classify the following measures as scalars and vectors.
10–19 coulomb

Classify the following measures as scalars and vectors.
20 m/s2

Classify the following measures as scalars and vectors.
time period

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation