Supplementary examples and questions from CBSE

Chapter 10 Class 12 Vector Algebra
Serial order wise

### Transcript

Supplementary Exercise Q5 Find the value of ๐ if the following vectors are co-planar (i) ๐ โ = 2๐ ฬ โ ๐ ฬ + ๐ ฬ, ๐ โ = ๐ ฬ + 2๐ ฬ โ 3๐ ฬ, and ๐ โ = 3๐ ฬ โ ๐๐ ฬ + 5๐ ฬ Three vectors ๐ โ, ๐ โ, ๐ โ are coplanar if [๐ โ" " ๐ โ" " ๐ โ ] = 0 Given, ๐ โ = 2๐ ฬ โ ๐ ฬ + ๐ ฬ ๐ โ = ๐ ฬ + 2๐ ฬ โ 3๐ ฬ ๐ โ = 3๐ ฬ โ ฮป ๐ ฬ + 5๐ ฬ [๐ โ" " ๐ โ" " ๐ โ ] = |โ 8(2&โ1&1@1&2&โ3@3&โ๐&5)| 0 = 2[(2ร5)โ(โฮปรโ3) ] โ (โ1) [(1ร5)โ(3รโ3) ] + 1[(1รโฮป)โ(3ร2) ] 0 = 2[10โ3ฮป] + [5+9 ] + [โฮปโ6] 0 = 20 โ 6ฮป + 14 โ ฮป +โ 6 0 = โ7ฮป + 28 7ฮป = 28 ฮป = 28/7 ๐ = 4 Therefore, ๐ โ,๐,๐ โ are coplanar if ฮป = 4 Supplementary Exercise Q5 Find the value of ๐ if the following vectors are co-planar (ii) ๐ โ = 2๐ ฬ + ๐ ฬ + ๐ ฬ, ๐ โ = 2๐ ฬ โ ๐๐ ฬ + ๐ ฬ, and ๐ โ = 5๐ ฬ โ ๐ ฬ โ 3๐ ฬ Three vectors ๐ โ, ๐ โ, ๐ โ are coplanar if [๐ โ" " ๐ โ" " ๐ โ ] = 0 Given, ๐ โ = 2๐ ฬ + ๐ ฬ + ๐ ฬ ๐ โ = 2๐ ฬ โ ฮป๐ ฬ + ๐ ฬ ๐ โ = 5๐ ฬ + ๐ ฬ โ 3๐ ฬ [๐ โ" " ๐ โ" " ๐ โ ] = |โ 8(2&1&1@2&โ๐&1@5&1&โ3)| 0 = 2[(โฮปรโ3)โ(1ร1)] โ 1[(2รโ3)โ(5ร1)] + 1[(2ร1)โ(5รโฮป) ] 0 = 2[3ฮปโ1] โ 1[โ6โ5] + 1[2+5ฮป] 0 = 6ฮป โ 2 + 11 + 2 + 5ฮป 0 = 11ฮป + 11 โ11ฮป = 11 ฮป = 11/(โ11) ๐ = โ1 Therefore, ๐ โ,๐,๐ โ are coplanar if ฮป = โ1