+91-8076753736 support@wiredfaculty.com
logo
logo

Sponsor Area

Vector Algebra

Question
CBSEENMA12034110
Wired Faculty App

If straight a with rightwards arrow on top makes equal angles with straight i with hat on top comma space straight j with hat on top space and space straight k with hat on top and has magnitude 3, then prove that the angle between straight a with rightwards arrow on top and each of straight i with hat on top comma space straight j with hat on top space and space straight k with hat on top is cos to the power of negative 1 end exponent open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses.

Solution

Let straight a with rightwards arrow on top space equals 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 such that open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 3 and O be the angle which straight a with rightwards arrow on top makes with straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top.
therefore space space space space cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top. space straight i with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight i with hat on top close vertical bar end fraction space equals space fraction numerator straight a subscript 1 over denominator left parenthesis 3 right parenthesis left parenthesis 1 right parenthesis end fraction space space space rightwards double arrow space space space space straight a subscript 1 space equals space 3 space cos space straight theta
Again cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top. space straight j with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight j with hat on top close vertical bar end fraction space equals space fraction numerator straight a subscript 2 over denominator left parenthesis 3 right parenthesis thin space left parenthesis 1 right parenthesis end fraction space space space rightwards double arrow space space straight a subscript 2 space equals space 3 space cos space straight theta
and cos space straight theta space equals space fraction numerator straight a with rightwards arrow on top. space straight k with hat on top over denominator open vertical bar straight a with rightwards arrow on top close vertical bar. space open vertical bar straight j with hat on top close vertical bar end fraction space equals space fraction numerator straight a subscript 3 over denominator left parenthesis 3 right parenthesis thin space left parenthesis 1 right parenthesis space end fraction space space rightwards double arrow space space straight a subscript 3 space equals space 3 space cos space straight theta
Also, open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 3                rightwards double arrow space space space space space square root of straight a subscript 1 squared plus straight a subscript 2 squared plus straight a subscript 3 squared end root space equals space 3
rightwards double arrow space space space square root of 9 space cos squared straight theta space plus space 9 space cos squared straight theta space plus space 9 space cos squared space straight theta end root space equals space 3 space space space space space space space rightwards double arrow space space space square root of 27 space cos squared straight theta end root space equals space 3 rightwards double arrow space space space 3 square root of 3 space cos space straight theta space equals space 3 space space space space space space rightwards double arrow space space space cos space straight theta space equals space fraction numerator 1 over denominator square root of 3 end fraction therefore space space space space space space space straight theta space space equals space cos to the power of negative 1 end exponent space open parentheses fraction numerator 1 over denominator square root of 3 end fraction close parentheses

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