HOW TO SHOW THE GIVEN VECTOR 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| = √(2² + (-1)² + 1²) = √6

|BC vector| = √(3² + (-4)² + (-4)² = √41

|CA vector| = √(1² + (-3)² + (-5)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 + 4²) = √21

|BC vector| = √((-1)² + 3² + (-5)²) = √35

|CA vector| = √(3² + (-2)² + 1²) = √14

|BC|² = |AB|² + |CA|²

Since the Pythagorean Theorem is satisfied, the given points form a right triangle.

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