Find the vector equation of the plane through the points A(3,-5,-1); B

1.	Find the vector equation of the plane through the points A(3,-5,-1); B(-1,5,7) and parallel to the vector 3i - j + 7k.
Answer» The position vector of given points A and B are respectively (3i - j + 7k and -i + 5j + 7k) Let P be any point on the plane with position vector xi + 5j + 2k So, vector AP = (x - 3)i + (y + 5)j + (z + 1)k and AB = -4i + 10j + 8k Clearly, vector(AP, AB) and given vector 3i - j + 7k are co-planar, So, equation of the require plane is \|(x - 3,y + 5,z + 1),(-4,10,8),(3,-1,7)\| = 0 78(x - 3) + 52(y + 5) - 26(z + 1) = 0 78x + 52y - 26z = 0 3x + 2y - z = 0 The vector equation of require plane is vector r.(3i + 2j - k) = 0

Find the vector equation of the plane through the points A(3,-5,-1); B(-1,5,7) and parallel to the vector 3i - j + 7k.

Answer»

The position vector of given points A and B are respectively (3i - j + 7k and -i + 5j + 7k)

Let P be any point on the plane with position vector xi + 5j + 2k

So, vector AP = (x - 3)i + (y + 5)j + (z + 1)k and AB = -4i + 10j + 8k

Clearly, vector(AP, AB) and given vector 3i - j + 7k are co-planar,

So, equation of the require plane is

|(x - 3,y + 5,z + 1),(-4,10,8),(3,-1,7)| = 0

78(x - 3) + 52(y + 5) - 26(z + 1) = 0

78x + 52y - 26z = 0

3x + 2y - z = 0

The vector equation of require plane is

vector r.(3i + 2j - k) = 0