PROPERTIES OF COPLANAR VECTORS

Problem 1 :

Show that the vectors

are coplanar.

Solution :

So, the given vectors are coplanar.

Problem 2 :

If 

are coplanar. Find the value of m.

Solution :

Problem 3 :

Show that the four points (6, -7, 0), (16, -19, -4), (0, 3, -6), (2, -5,10)  lie on a same plane.

Solution :

Let the points  A = (6, -7, 0), B = (16, -19, -4), C = (0, 3, -6), D = (2, -5,10).

To show that the four points A, B, C and D lie on the plane, we have to prove that the three vectors 

Finding AB vector :

Finding AC vector :

Finding AD vector :

Proving the given four vectors are coplanar :

Problem 4 :

If the vectors

are coplanar, then prove that the vectors

are also coplanar.

Solution :

Since the vectors 

are coplanar, then 

Using the property of scalar triple product, we get

Problem 5 :

Determine whether the three vectors

are coplanar.

Solution :

Problem 6 :

Let 

If c₁ = 1 and c₂ = 2, find c₃ such that 

are coplanar.

Solution :

After applying the values of c₁ and c₂, we get

Problem 7 :

If 

show that 

depends on neither x nor y.

Solution :

Problem 8 :

If the vectors 

are coplanar, prove that c is the geometric mean of 

Solution :

Since the vectors are coplanar,

Problem 9 :

Let

the three non zero vectors such that c vector is a unit perpendicular to both a vector and c vector. If the angle between a vector and b vector is π/6, show that

Solution :

c vector is perpendicular to both a vector and b vector. So, c vector is parallel to (a vector x b vector)

Squaring on both sides,