(I) The vector equation of the straight line passing through the point A (a) and parallel to the vector b is

r = a + t b, t ∈ R

Proof:

Let OP = r be any point on the line passing through OA = a.

Let b be any vector parallel to AP.

∴ AP and b are collinear vectors.

∴ AP = t b, t ∈ R (by the definition of collinear vectors)

⇒ OP – OA = t b

⇒ OP = OA + t b

∴ r = a + t b, t ∈ R

II) The vector equation of the line passing through the points A (a) and B (b) is

r = (1 – t) a + t b, t ∈ R.

Proof:

Let OP = r be point of the line passing through the points OA = a and OB = b

⇔ AP and AB are collinear vectors.

⇔ AP = t AB, t ∈ R

⇔ OP – OA = t (OB – OA)

⇔ OP = OA + t OB – t OA

⇔ OP = (1 – t) OA + t OB

⇔ r = (1 – t) a + t b, t ∈ R

(III) The vector equation of the plane passing through the point A (a) and parallel to the vectors b and c is

r = a + t b + s c, t, s ∈ R

Proof :

Let OP = r be a point on the plane passing through the points OA = a, and parallel to the vectors b and c as shown in figure.

In a plane, through the point A, draw lines parallel to the vectors b and c with line segment AP as diagonal.

Complete the parallelogram ALPM in the a plane.

∴ A L = s c, s ∈ R and AM = t b, t ∈ R

(IV) The vector equation of the plane passing through the points A(a), B(b) and parallel to the vector c is

r = (1 – t) a + t b + s c where t, s ∈ R

Proof:

We know that "The vector equation of the plane passing through the point A(a) and parallel to the vector b and c is

r = a + t b + s c where t, s ∈ R"

Replacing the vector b with AB, the equation of the plane is

r = a + t AB + s c

r = a + t (OB – OA) + sc

r = a + t b – t a + sc (∵ OA = a, OB = b)

∴ r = (1 – t)a + t b + s c

(V) The vector equation of the plane passing through three non-collinear points A(a), B(b), and C(c) is

r = (1 – t – s)a + t b + sc where t, s ∈ R

Proof:

Knowing that "The vector equation of the plane passing through the point A(a) and parallel to the vectors b and c is

r = a + t b + s c where t, s ∈ R"

Replacing the vector b with AB and c with AC, the equation of the plane is