Check sibling questions

Show that the vectors 2i - 3j + 4k and -4i + 6j - 8k are collinear

Ex 10.2, 11 - Chapter 10 Class 12 Vector Algebra - Part 2
Ex 10.2, 11 - Chapter 10 Class 12 Vector Algebra - Part 3

This video is only available for Teachoo black users

Solve all your doubts with Teachoo Black (new monthly pack available now!)


Transcript

Ex 10.2, 11 (Method 1) Show that the vectors 2𝑖 ̂ − 3𝑗 ̂ + 4𝑘 ̂ and − 4𝑖 ̂ + 6 𝑗 ̂ − 8𝑘 ̂ are collinear.Two vectors are collinear if they are parallel to the same line. Let 𝑎 ⃗ = 2𝑖 ̂ − 3𝑗 ̂ + 4𝑘 ̂ and 𝑏 ⃗ = –4𝑖 ̂ + 6𝑗 ̂ – 8𝑘 ̂ Magnitude of 𝑎 ⃗ = √(22+(−3)2+42) |𝑎 ⃗ | = √(4+9+16) = √29 Directions cosines of 𝑎 ⃗ = (2/√29,(−3)/√29,4/√29) Magnitude of 𝑏 ⃗ =√((−4)2+62+(−8)2) |𝑏 ⃗ | = √(16+36+64) = √116 = 2√29 Directions cosines of 𝑏 ⃗ = ((−4)/(2√29),6/(2√29),(−8)/(2√29)) = ((−2)/√29,3/√29,(−4)/√29) = −1(2/√29,(−3)/√29,4/√29) Hence, Direction cosines of 𝒂 ⃗ = (−1) × Direction cosines of 𝒃 ⃗ ∴ They have opposite directions Since 𝑎 ⃗ and 𝑏 ⃗ are parallel to the same line 𝑚 ⃗, they are collinear. Hence proved Ex 10.2, 11 (Method 2) Show that the vectors 2𝑖 ̂ − 3𝑗 ̂ + 4𝑘 ̂ and − 4𝑖 ̂ + 6 𝑗 ̂ − 8𝑘 ̂ are collinear.𝑎 ⃗ = 2𝑖 ̂ − 3𝑗 ̂ + 4𝑘 ̂ 𝑏 ⃗ = –4𝑖 ̂ + 6𝑗 ̂ – 8𝑘 ̂ Two vectors are collinear if their directions ratios are proportional 𝑎_1/𝑏_1 = 𝑎_2/𝑏_2 = 𝑏_3/𝑏_3 = 𝜆 2/(−4) = (−3)/6 = 4/(−8) = (−1)/2 Since, directions ratios are proportional Hence, 𝑎 ⃗ & 𝑏 ⃗ are collinear

Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.