# HOW TO SHOW THE GIVEN VECTORS FORM RIGHT TRIANGLE

## About "How to Show the Given Vectors Form Right Triangle"

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  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 = 02

|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

52 + 62 + 72 + 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

After having gone through the stuff given above, we hope that the students would have understood,"How to Show the Given Vectors Form Right Triangle"

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