Supplementary Example 5

Transcript

Supplementary Example 5 Show that the four points with position vectors 6 𝑖﷯ − 7 𝑗﷯, 16 𝑖﷯ − 19 𝑗﷯ − 4 𝑘﷯, 3 𝑗﷯ − 6 𝑘﷯ and 2 𝑖﷯ + 5 𝑗﷯ + 10 𝑘﷯ are not co-planar Let points be A = 6 𝑖﷯ – 7 𝑗﷯ B = 16 𝑖﷯ – 19 𝑗﷯ – 4 𝑘﷯ C = 3 𝑗﷯ – 6 𝑘﷯ D = 2 𝑖﷯ + 5 𝑗﷯ +10 𝑘﷯ Four points A, B, C, D are coplanar if the three vectors 𝐴𝐵﷯ , 𝐴𝐶﷯ and 𝐴𝐷﷯ are coplanar. i.e. 𝑨𝑩﷯, 𝑨𝑪﷯, 𝑨𝑫﷯﷯ = 0 = 10 100+72﷯+12 −60−24﷯−4 −72+40﷯ = 10 172﷯+12 −84﷯−4 −32﷯ = 1720 – 1008 + 128 = 840 ∴ 𝑨𝑩﷯, 𝑨𝑪﷯, 𝑨𝑫﷯﷯ ≠ 0 Therefore, points A, B, C and D are not coplanar.