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

Vector Algebra

Question
CBSEENMA12034215
Wired Faculty App

The two adjacent sides of a parallelogram are 2 straight i with hat on top minus 4 straight j with hat on top plus 5 straight k with hat on top and straight i with hat on top space minus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top.
Find the unit vector parallel to its diagonal. Also find its area.

Solution
Let ABCD be the parallelogram with
                 AB with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top space and space AD with rightwards arrow on top space equals straight i with hat on top minus 2 space straight j with hat on top minus 3 space straight k with hat on top
Now   AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space AD with rightwards arrow on top
                      equals left parenthesis 2 space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis plus left parenthesis straight i with hat on top minus 2 space straight j with hat on top space minus space 3 space straight k with hat on top right parenthesis space equals space 3 space straight i with hat on top space minus space 6 space straight j with hat on top space plus space 2 space straight k with hat on top
Here a unit vector parallel to 
AC with rightwards arrow on top space equals space fraction numerator 1 over denominator open vertical bar AC with rightwards arrow on top close vertical bar end fraction left parenthesis stack AC right parenthesis with rightwards arrow on top space space space space equals space fraction numerator 1 over denominator square root of 3 squared plus left parenthesis negative 6 right parenthesis squared plus 2 squared end root end fraction left parenthesis 3 straight i with hat on top minus 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space space equals space fraction numerator 1 over denominator square root of 9 plus 36 plus 4 end root end fraction left parenthesis 3 space straight i with hat on top space minus space 6 space straight j with hat on top space space plus space 2 space straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 49 end fraction left parenthesis 3 straight i with hat on top space minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals space 1 over 7 left parenthesis 3 straight i with hat on top space minus space 6 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis space equals space 3 over 7 straight i with hat on top space minus space 6 over 7 straight j with hat on top space plus space 2 over 7 straight k with hat on top
Vector area of parallelogram ABCD = AB with rightwards arrow on top space cross times space AD with rightwards arrow on top
                     equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell space space straight k with hat on top end cell row 2 cell negative 4 end cell cell space space 5 end cell row 1 cell negative 2 end cell cell negative 3 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 4 end cell cell space space space space 5 end cell row cell negative 2 end cell cell space minus 3 end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 2 5 row 1 cell negative 3 end cell end table close vertical bar space plus space straight k with hat on top space open vertical bar table row 2 cell negative 4 end cell row 1 cell negative 2 end cell end table close vertical bar equals space straight i with hat on top left parenthesis 12 plus 10 right parenthesis space minus space straight j with hat on top left parenthesis 6 minus 5 right parenthesis space plus space straight k with hat on top left parenthesis negative 4 plus 4 right parenthesis equals space 22 straight i with hat on top space plus 11 space straight j with hat on top space plus space 0 space straight k with hat on top therefore space area space of space parallelogram space equals space open vertical bar AB with rightwards arrow on top space cross times AD with rightwards arrow on top close vertical bar space equals space open vertical bar 22 space straight i with hat on top space plus space 11 space straight j with hat on top space plus space 0 space straight k with hat on top close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals square root of left parenthesis 22 right parenthesis squared plus left parenthesis 11 right parenthesis squared plus 0 squared end root space equals space square root of 484 plus 121 plus 0 end root space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space square root of 605 space equals space square root of 21 space cross times 5 end root space space equals space 11 square root of 5 space space sq. space units.

Some More Questions From Vector Algebra Chapter

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

Classify the following measures as scalars and vectors.
distance

Classify the following measures as scalars and vectors.
force

Classify the following measures as scalars and vectors.
velocity

Classify the following measures as scalars and vectors.
work done

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation