Supplementary Exercise Q7

Transcript

Supplementary Exercise Q7 Show that the four points A, B, C and D with position vectors 4 𝑖﷯ + 5 𝑗﷯ + 𝑘﷯, −( 𝑗﷯ + 𝑘﷯), 3 𝑖﷯ + 9 𝑗﷯ + 4 𝑘﷯ and −4 𝑖﷯ + 4 𝑗﷯ + 4 𝑘﷯, 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 𝐴𝐵﷯, 𝐴𝐶﷯, 𝐴𝐷﷯﷯ = −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.