vector(4a - 3b - c), vector(3a+7b - 10c), vector(2a+5b - 7c), vector(a+ 2b - 3c) with position vectors are coplanar and find the equation of the plane

Question

Show that the points vector(4a - 3b - c), vector(3a+7b - 10c), vector(2a+5b - 7c), vector(a+ 2b - 3c) with position vectors are coplanar and find the equation of the plane through them

Answer 1

Let A, B, C, D be the given points respectively

vector{OA = 4a - 3b - c},

vector{OB = 3a + 7b - 10c}, 

vector{OC = 2a + 5b - 7c}, 

vector{OD = a2b - 3c}

vector{AB = OB - OA = (3a + 7b - 10c) - (4a - 3b - c) = -a + 10b - 9c}

vector{AC = OC - OA = (2a + 5b - 7c) - (4a - 3b - c) = -2a + 8b - 6c}

vector{AD = OD - OA = (a + 2b - 3c) - (4a - 3b - c) = -3a + 5b - 2c}

Let vector{xAB + yAC = AD}

Then x( -a + 10b - 9c) + y( -2a + 8b - 6c) = -3a + 5b - 2c

=> (–x – 2y)a + (10x + 8y) b + (–9x – 6y) c = -3a + 5b - 2c

=> –x – 2y = –3  (1), 

=> 10x + 8y = 5  (2), 

=> –9x – 6y = –2  (3)

4(1) + (2) => 6x = –7 => x = –7/6 

(1) => 7/6 – 2y = –3 => 2y = 7/6 + 3 = 25/6 => y = 25/12 

Also – 9(–7/6) – 6(25/12) = 21/2 – 25/2 = –4/2 = –2

x = –7/6, y = 25/12 satisfy the equations (1), (2) and (3) 

vector{AB,AC,AD} are coplanar

A, B, C, D are coplanar. 

The vector equation of the plant passing through the points

vector{4a - 3b - c, 3a + 7b - 10c, 2a + 5b - 7c, a + 2b - 3c} is 

vector{r = [(1 - s - t)(4a - 3b - c) + s(3a + 7b - 10c) + t(2a + 5b - 7c)]}