HCJC Year 2000 Paper 1 Q11
Given that a = 3i + j + k, b = i - k, c = 4i - 3j + 2k are position vectors of the points A, B and C respectively, find the position vectors of the points and Q which divide BC internally and externally in the ratio 1:2, respectively.
Show that AP is perpendicular to BC.
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Answer:
By Ratio Theorem,
OP = ⅓(OC + 2OB)
OP = ⅓(4i - 3j + 2k + 2i - 2k)
OP = 2i - j
By Ratio Theorem,
OB = ½(OQ + OC)
OQ = 2OB - OC
OQ = 2i - 2k - (4j - 3j + 2k)
OQ = -2i + 3j -4k
AP = OP - OA = -i - 2j - k
BC = OC - OB = 3i - 3j + 3k
AP · BC = -3 + 6 - 3 = 0
Hence, AP is perpendicular to BC (shown)
