**How to Show the Given Vectors Form Right Triangle :**

Here we are going to see how to show the given vectors form a right triangle.

To show the given vectors form a right triangle, we have to find the dot product of two vectors.

If two vectors a vector and b vector are perpendicular , then a vector . b vector = 0

**Question 1 :**

Show that the vectors a = 2i + 3j + 6k, b = 6i + 2j − 3k, and c = 3i − 6j + 2k are mutually orthogonal.

**Solution :**

Mutually orthogonal means, they are perpendicular to each other.

a vector . b vector = (2i + 3j + 6k) . (6i + 2j − 3k)

= 2(6) + 3(2) + 6(-3)

= 12 + 6 - 18

= 18 - 18

a . b = 0

b vector . c vector = (6i + 2j − 3k) . (3i − 6j + 2k)

= 6(3) + 2(-6) + (-3)2

= 18 - 12 - 6

= 18 - 18

b . c = 0

c vector . a vector = (3i − 6j + 2k) . (2i + 3j + 6k)

= 3(2) + (-6)(3) + 2(6)

= 6 - 18 + 12

= 18 - 18

c . a = 0

Hence the given vectors are mutually orthogonal.

**Question 2 :**

Show that the vectors −i − 2j − 6k, 2i − j + k, and − i + 3j + 5k form a right angled triangle.

**Solution :**

Let a vector = −i − 2j − 6k, b vector = 2i − j + k and c vector = − i + 3j + 5k

a vector . b vector = (−i − 2j − 6k) . (2i − j + k)

= -1(2) + 2(-1) + (-6)(1)

= -2 - 2 - 6

= -10

b vector . c vector = (2i − j + k) . (− i + 3j + 5k)

= 2(-1) + (-1)(3) + 1(5)

= -2 - 3 + 5

= 0

b vector and c vector are perpendicular.

Hence the given vectors form a right triangle.

**Question 3 :**

If | a vector| = 5, | b vector| = 6, | c vector | = 7 and a vector + b vector + c vector = 0, find a ⋅ b + b ⋅ c + c ⋅ a .

**Solution :**

a vector + b vector + c vector = 0

|a vector + b vector + c vector| = 0

Taking squares on both sides, we get

|a vector + b vector + c vector|^{2} = 0^{2}

|a|^{2} + |b|^{2} + |c|^{2} + 2|a||b| + 2|b||c| + 2|c||a| = 0

|a|^{2} + |b|^{2} + |c|^{2} + 2[a.b + b.c + c.a] = 0

Applying the value of |a|, |b| and |c|, we get

5^{2} + 6^{2} + 7^{2} + 2[a.b + b.c + c.a] = 0

(25 + 36 + 49) + 2[a.b + b.c + c.a] = 0

2[a.b + b.c + c.a] = -110

a.b + b.c + c.a = -110/2 = -55

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