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

Vector Algebra

Question
CBSEENMA12034262
Wired Faculty App

If with reference to a right handed system of mutually perpendicular unit vector unit vector straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top

express straight beta with rightwards arrow on top in the form straight beta with rightwards arrow on top in the form straight beta with rightwards arrow on top space equals space straight beta with rightwards arrow on top subscript 1 space equals space straight beta with rightwards arrow on top subscript 1 space plus space straight beta with rightwards arrow on top subscript 2 space end subscript space where space stack straight beta subscript 1 with rightwards arrow on top space is space parallel space to space straight alpha space and space stack straight beta subscript 2 with rightwards arrow on top space is space perpendicular space to space straight alpha with rightwards arrow on top.

Solution

Let straight alpha with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space straight j with hat on top comma space space space space straight beta with rightwards arrow on top space equals space 2 straight i with hat on top plus straight j with hat on top space minus space 3 straight k with hat on top
Let  stack straight beta subscript 1 with rightwards arrow on top space equals space straight a subscript 1 straight i with hat on top space plus space straight a subscript 2 straight j with hat on top space plus space straight a subscript 3 straight k with hat on top comma space space stack straight beta subscript 2 with rightwards arrow on top space equals space straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top
therefore space space space space space straight beta with rightwards arrow on top space equals straight beta with rightwards arrow on top subscript 1 plus space stack straight beta subscript 2 with rightwards arrow on top space equals space left parenthesis straight a subscript 1 plus straight b subscript 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight a subscript 2 plus straight b subscript 2 right parenthesis straight j with hat on top space plus space left parenthesis straight a subscript 3 plus straight b subscript 3 right parenthesis straight k with hat on top rightwards double arrow space space 2 straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top space equals space left parenthesis straight a subscript 1 plus straight b subscript 1 right parenthesis straight i with hat on top space plus space left parenthesis straight a subscript 2 plus straight b subscript 2 right parenthesis space straight j with hat on top space plus space left parenthesis straight a subscript 3 plus straight b subscript 3 right parenthesis space straight k with hat on top rightwards double arrow space space space straight a subscript 1 plus straight b subscript 1 space equals space 2 comma space space space space space straight a subscript 2 plus straight b subscript 2 space equals space 1 comma space space straight a subscript 3 plus straight b subscript 3 space equals space minus 3 therefore space space space space space space straight b subscript 1 space equals space 2 space minus straight a subscript 1 comma space space space space space straight b subscript 2 space equals space 1 minus straight a subscript 2 comma space space space straight b subscript 3 space equals space minus 3 minus straight a subscript 3 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... left parenthesis 1 right parenthesis space space space space space space space end subscript
Since stack straight beta subscript 1 with rightwards arrow on top is parallel to straight alpha with rightwards arrow on top
 therefore space space space space space space straight beta with rightwards arrow on top subscript 1 space cross times space straight alpha with rightwards arrow on top space equals space 0 space space space space space space rightwards double arrow space space space space 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 straight k with hat on top end cell row cell straight a subscript 1 end cell cell straight a subscript 2 end cell cell straight a subscript 3 end cell row 3 cell negative 1 end cell 0 end table close vertical bar space equals space 0 with rightwards arrow on top rightwards double arrow space space space space space space straight a subscript 3 straight i with hat on top space plus space 3 space straight a subscript 3 straight j with hat on top space plus left parenthesis negative straight a subscript 1 space minus space 3 space straight a subscript 2 right parenthesis space straight k with hat on top space equals space 0 with rightwards arrow on top rightwards double arrow space space space space space space straight a subscript 3 space equals space 0 comma space space space space 3 space straight a subscript 3 space equals space 0 comma space space space space space minus straight a subscript 1 space minus space 3 space straight a subscript 2 space equals space 0 rightwards double arrow space space space space space straight a subscript 3 space equals space 0 space space space straight a subscript 1 space equals space minus 3 space straight a subscript 2 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 space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis Again comma space space stack straight beta subscript 2 with rightwards arrow on top space is space perpendicular space to space straight alpha with rightwards arrow on top therefore space space space space space stack straight beta subscript 2 with rightwards arrow on top space. space straight alpha with rightwards arrow on top space equals space 0 space space space space space space rightwards double arrow space space space space left parenthesis straight b subscript 1 straight i with hat on top space plus space straight b subscript 2 straight j with hat on top space plus space straight b subscript 3 straight k with hat on top right parenthesis. space space space left parenthesis 3 space straight i with hat on top space minus space straight j with hat on top right parenthesis space equals space 0 rightwards double arrow space space space space 3 space straight b subscript 1 space minus space straight b subscript 2 space equals space 0 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 space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 1 right parenthesis close square brackets rightwards double arrow space 3 space left parenthesis 2 minus straight a subscript 1 right parenthesis space minus space 1 left parenthesis 1 minus straight a subscript 2 right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space space space 6 minus 3 space straight a subscript 1 space minus space 1 space plus space straight a subscript 2 space equals space 0 rightwards double arrow space space space 5 minus 3 space straight a subscript 1 space plus space straight a subscript 2 space equals space 0 space space space space space space rightwards double arrow space space space 5 plus 9 space straight a subscript 2 plus space straight a subscript 2 space space space equals space 0 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 space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 2 right parenthesis close square brackets rightwards double arrow space space space space space 10 space straight a subscript 2 space equals space minus 5 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 rightwards double arrow space space space space space space straight a subscript 2 space equals space minus 1 half 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 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 space space space space space space
therefore space space space space space straight a subscript 1 space equals 3 over 2                                                                               open square brackets because space of space left parenthesis 1 right parenthesis close square brackets
therefore space space space space straight beta with rightwards arrow on top subscript 1 space equals space 3 over 2 straight i with hat on top space minus space 1 half straight j with hat on top
From (1),   straight b subscript 1 space equals 2 minus 3 over 2 space equals space 1 half comma space space space space space space space straight b subscript 2 equals space 1 plus 1 half space equals 3 over 2 comma space space straight b subscript 3 space equals space minus 3 minus 0 space equals space minus 3
therefore space space stack straight beta subscript 2 with rightwards arrow on top space equals space 1 half straight i with hat on top space plus space 3 over 2 straight j with hat on top space minus space 3 space straight k with hat on top.
 

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

Classify the following measures as scalars and vectors.
distance

Classify the following measures as scalars and vectors.
force

Mock Test Series

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation