Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Supplementary examples and questions from CBSE
Supplementary Example 2 Important Deleted for CBSE Board 2024 Exams
Supplementary Example 3 Deleted for CBSE Board 2024 Exams
Supplementary Example 4 Deleted for CBSE Board 2024 Exams
Supplementary Example 5 Important Deleted for CBSE Board 2024 Exams
Supplementary Example 6 Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q1 Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q2 Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q3 Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q4 Important Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q5 Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q6 Important Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q7 Deleted for CBSE Board 2024 Exams You are here
Supplementary Exercise Q8 Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q9 Important Deleted for CBSE Board 2024 Exams
Supplementary Exercise Q10 Deleted for CBSE Board 2024 Exams
Supplementary examples and questions from CBSE
Last updated at May 29, 2023 by Teachoo
Supplementary Exercise Q7 Show that the four points A, B, C and D with position vectors 4𝑖 ̂ + 5𝑗 ̂ + 𝑘 ̂, −(𝑗 ̂ + 𝑘 ̂), 3𝑖 ̂ + 9𝑗 ̂ + 4𝑘 ̂ and −4𝑖 ̂ + 4𝑗 ̂ + 4𝑘 ̂, respectively are co-planar Four points A, B, C, D are coplanar if the three vectors (𝐴𝐵) ⃗ , (𝐴𝐶) ⃗ and (𝐴𝐷) ⃗ are coplanar. i.e. [(𝑨𝑩) ⃗, (𝑨𝑪) ⃗, (𝑨𝑫) ⃗ ] = 0 A (4𝑖 ̂ + 5𝑗 ̂ + 𝑘 ̂) B (−𝑗 ̂ − 𝑘 ̂) (𝑨𝑩) ⃗ = (0𝑖 ̂ − 𝑗 ̂ − 𝑘 ̂) − (4𝑖 ̂ + 5𝑗 ̂ + 𝑘 ̂) = −4𝒊 ̂ − 6𝒋 ̂ − 2𝒌 ̂ A (4𝑖 ̂ + 5𝑗 ̂ + 𝑘 ̂) C (3𝑖 ̂ + 9𝑗 ̂ + 4𝑘 ̂) (𝑨𝑪) ⃗ = (3𝑖 ̂ + 9𝑗 ̂ + 4𝑘 ̂) − (4𝑖 ̂ + 5𝑗 ̂ + 𝑘 ̂) = –𝒊 ̂ + 4𝒋 ̂ + 3𝒌 ̂ A (4𝑖 ̂ + 5𝑗 ̂ + 𝑘 ̂) D (−4𝑖 ̂ + 4𝑗 ̂ + 4𝑘 ̂) (𝑨𝑫) ⃗ = (−4𝑖 ̂ + 4𝑗 ̂ + 4𝑘 ̂) − (4𝑖 ̂ + 5𝑗 ̂ + 𝑘 ̂) = –8𝒊 ̂ − 𝒋 ̂ + 3𝒌 ̂ [(𝐴𝐵) ⃗, (𝐴𝐶) ⃗, (𝐴𝐷) ⃗ ] = |■8(−4&−6&−2@−1&4&3@−8&−1&3)| = −4[(4×3)−(−1×3) ] − (−6) [(–1 × 3) – (–8 × 3)] + (−2)[(−1×−1)−(−8×4) ] = –4 [12+3]+6[−3+24]−2[1+32] = −4 (15) + 6 (21) − 2 (33) = −60 + 126 − 66 = −126+ 126 = 0 ∴[(𝐴𝐵) ⃗, (𝐴𝐶) ⃗, (𝐴𝐷) ⃗ ] = 0 Therefore, points A, B, C and D are coplanar.