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प्रश्न
If A, B, C are three non- collinear points with position vectors `vec a, vec b, vec c`, respectively, then show that the length of the perpendicular from Con AB is `|(vec a xx vec b)+(vec b xx vec c) + (vec b xx vec a)|/|(vec b - vec a)|`
उत्तर
Let ABC be a triangle and let `vec a, vec b, vec c` be the position vectors of its vertices A, B, C respectively. Let CM be the perpendicular from C on AB. Then
Area of ΔABC = `1/2 (AB).CM = 1/2 |vec(AB)|(CM)`
Also Area of ΔABC = `1/2 |vec(AB) xx vec(AC)|`
Area of ΔABC = `1/2 |veca xx vecb + vecbxxvecc + vecc xx vec a|`
`:. 1/2|vec(AB)|(CM) = 1/2|vec a xx vec b + vec b xx vec c + vec c xx vec a|`
`=> CM = |vec a xx vec b + vec b xx vec c + vec c xx vec a|/|vec(AB)|`
`=> CM = |vec a xx vec b + vec b xx vec c + vec c xx vec a|/|vec b - vec a|`
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