In a right triangle, By Pythagorean Theorem, square of the longest side (hypotenuse) is always equal to the sum of the squares of other two legs.
In vector algebra, to prove the given vectors form a right triangle, find the vectors which represent sides of the triangle using the given position vectors (if required).
AB vector = OB vector - OA vector
BC vector = OC vector - OB vector
CA vector = OA vector - OC vector
After having found the vectors for the sides, Pythagorean Theorem can be used to prove that they form a right triangle.
Example 1:
Show that the following vectors form a right triangle :
2i - j + k
3i - 4j - 4k
i - 3j - 5k
Solution :
Let
AB vector = 2i - j + k
BC vector = 3i - 4j - 4k
CA vector = i - 3j - 5k
|AB vector| = √(2^{2} + (-1)^{2} + 1^{2}) = √6
|BC vector| = √(3^{2} + (-4)^{2} + (-4)^{2} = √41
|CA vector| = √(1^{2} + (-3)^{2} + (-5)^{2} = √35
|AB|^{2} = (√6)^{2}
|AB|^{2} = 6
|BC|^{2} = (√41)^{2}
|BC|^{2} = 41
|CA|^{2} = (√35)^{2}
|CA|^{2} = 35
|BC|^{2} = |AB|^{2} + |CA|^{2}
Since the Pythagorean Theorem is satisfied, the given vectors forma right triangle.
Example 2:
Show that the points whose position vectors given below form a right triangle.
4i - 3j + k
2i - 4j + 5k
i - j
Solution :
Let the given points be A, B and C. Then
OA vector = 4i - 3j + k
OB vector = 2i - 4j + 5k
OC vector = i - j
AB vector :
AB vector = OB vector - OA vector
= (2i - 4j + 5k) - (4i - 3j + k)
= 2i - 4j + 5k - 4i + 3j - k
= -2i - j + 4k
BC vector :
BC vector = OC vector - OB vector
= (i - j) - (2i - 4j + 5k)
= i - j - 2i + 4j - 5k
= -i + 3j - 5k
CA vector :
CA vector = OA vector - OC vector
= (4i - 3j + k) - (i - j)
= 4i - 3j + k - i + j
= 3i - 2j + k
|AB vector| = √((-2)^{2} + (-1)^{2} + 4^{2}) = √21
|BC vector| = √((-1)^{2} + 3^{2} + (-5)^{2}) = √35
|CA vector| = √(3^{2} + (-2)^{2} + 1^{2}) = √14
|AB|^{2} = (√21)^{2}
|AB|^{2} = 21
|BC|^{2} = (√35)^{2}
|BC|^{2} = 35
|CA|^{2} = (√14)^{2}
|CA|^{2} = 14
|BC|^{2} = |AB|^{2} + |CA|^{2}
Since the Pythagorean Theorem is satisfied, the given points form a right triangle.
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