How to Show the Given Vector Form a Right Triangle :
Here we are going to see how to show the given vector form a right triangle.
Right triangle means, one of the angles must be 90 degrees. We may prove this using Pythagorean theorem. That is,
The length of square of longer side must be equal to the sum of the squares of lengths of the other two sides.
If we are given position vectors, we have to find the sides of the triangle (AB, BC and CA)
Here AB vector = OB vector - OA vector
BC vector = OC vector - OB vector
CA vector = OA vector - OC vector
After finding side lengths, we may apply Pythagorean theorem
Question 1 :
Show that the vectors 2i − j + k, 3i − 4j − 4k, i − 3j − 5k form a right angled triangle.
Here we have sides of the triangle ABC,
AB vector = 2i vector − j vector + k vector
BC vector = 3i vector − 4j vector - 4k vector
CA vector = i vector − 3j vector - 5k vector
|AB vector| = √(22 + (-1)2 + 12) = √6
|BC vector| = √(32 + (-4)2 + (-4)2 = √(9+16+16) = √41
|CA vector| = √(12 + (-3)2 + (-5)2 = √(1+9+25) = √35
|BC|2 = |AB|2 + |CA|2
(√41)2 = (√6)2 + (√35)2
41 = 6 + 35
41 = 41
It satisfies the condition.
Hence given vectors form a right triangle.
Question 2 :
Find the value of λ for which the vectors a = 3i + 2j + 9k and b = i + λj + 3k are parallel.
Since they are equal,
a vector = m b vector (here m is a constant)
3i + 2j + 9k = i + λj + 3k
3(i + (2/3)j + 3k) = i + λj + 3k
By equating the coefficients, we get λ = 2/3
Hence the value of λ is 2/3.
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