**Find the Area of a Parallelogram Formed by Vectors :**

Here we are going to see how to find the area of parallelogram formed by vectors.

**Question 1 :**

Find the area of the parallelogram whose two adjacent sides are determined by the vectors i vector + 2j vector + 3k vector and 3i vector − 2j vector + k vector.

**Solution :**

Let a vector = i vector + 2j vector + 3k vector

b vector = 3i vector − 2j vector + k vector.

Vector area of parallelogram = a vector x b vector

= i[2+6] - j[1-9] + k[-2-6]

= 8i + 8j - 8k

= √8^{2} + 8^{2} + (-8)^{2}

= √(64+64+64)

= √192

= 8√3

**Question 2 :**

Find the area of the triangle whose vertices are A(3, - 1, 2), B(1, - 1, - 3) and C(4, - 3, 1).

**Solution :**

A vector = (3i - j + 2k) vector

B vector = (i - j - 3k) vector

C vector = (4i - 3j + k) vector

AB vector = -2i vector -5k vector

AC vector = i vector - 2j vector - k vector

= i[0-10]-j[2+5]+k[4-0]

= -10i+7j-4k

Area of triangle = (1/2) |AB vector x AC vector|

= (1/2) √(-10)^{2} + 7^{2} + (-4)^{2}

= (1/2) √(100+49+16)

= (1/2) √165

Hence area of the given triangle is (1/2) √165.

**Question 3 :**

If a vector, b vector, c vector are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is (1/2) |a × b + b × c + c × a| vector. Also deduce the condition for collinearity of the points A, B, and C.

**Solution :**

Let OA vector = a vector

OB vector = b vector

OC vector = c vector

Area of triangle ABC = (1/2) |AB vector x AC vector|

= (1/2) |(OB - OA) x (OC - OA)|

= (1/2) |(b - a) x (c - a)|

= (1/2) |(b x c - b x a - a x c + a x a)|

= (1/2) |(b x c + a x b + c x a + 0 vector)|

= (1/2) |a x b + b x c + c x a|

If the points A, B and C are collinear, then

Area of triangle ABC = 0

(1/2) |a x b + b x c + c x a| = 0

|a x b + b x c + c x a| = 0

a x b + b x c + c x a = 0

Hence the required condition is a x b + b x c + c x a = 0.

After having gone through the stuff given above, we hope that the students would have understood, "Find the Area of a Parallelogram Formed by Vectors"

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