Let a and b be any two vectors.
Formula for dot product or scalar product of two vectors 'a' and 'b' :
a ⋅ b = |a||b|cos90°
We know that the value of of cos90° = 0.
a ⋅ b = |a||b|(0)
a ⋅ b = 0
From the above working, it is clear that if two vectors are perpendicular, then their dot product or scalar product is equal to zero. That is, if two vectors 'a' and 'b' are perpendicular, then
a ⋅ b = 0
Example 1 :
Check whether the vectors (4i + 2j - 8k) and (3i - 2j + k) are perpendicular to each other.
Solution :
Find the dot product of the given two vectors :
= (4i + 2j - 8k) ⋅ (3i - 2j + k)
= 4(3) + 2(-2) - 8(1)
= 12 - 4 - 8
= 0
Since the dot product is equal to zero, the given two vectors are perpendicular.
Example 2 :
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
= 0
b vector ⋅ c vector = (6i + 2j − 3k) ⋅ (3i − 6j + 2k)
= 6(3) + 2(-6) + (-3)2
= 18 - 12 - 6
= 18 - 18
= 0
c vector ⋅ a vector = (3i − 6j + 2k) ⋅ (2i + 3j + 6k)
= 3(2) + (-6)(3) + 2(6)
= 6 - 18 + 12
= 18 - 18
= 0
Since, a ⋅ b = b ⋅ c = c ⋅ a = 0, the given vectors are mutually orthogonal.
Example 3 :
Let -i - 2j - 6k, 2i - j + k, and -i + 3j + 5k be the sides of a triangle. Show that the above vectors form a right triangle.
Solution :
The given vectors are the sides of a right triangle. To show that the triangle is a right triangle, it is enough to prove that the two of its sides are perpendicular.
Let
a = -i - 2j - 6k
b = 2i - j + k
c = = -i + 3j + 5k
a ⋅ b = (-i - 2j - 6k) ⋅ (2i - j + k)
= -1(2) + (-2)(-1) + (-6)(1)
= -2 + 2 - 6
= -6 ≠ 0
b ⋅ c = (2i - j + k) ⋅ (-i + 3j + 5k)
= 2(-1) + (-1)(3) + 1(5)
= -2 - 3 + 5
= 0
Since b ⋅ c = 0, b vector and c vector are perpendicular.
So, the given vectors form a right triangle.
Example 4 :
If the vectors (i + 2j - 5k) and (3i - 2j + ak) are perpendicular to each other, find the value of 'a'.
Solution :
Since , the given two vectors are perpendicular, their dot product is equal to zero.
(i + 2j - 5k) ⋅ (3i - 2j + ak) = 0
1(3) + 2(-2) - 5(a) = 0
3 - 4 - 5a = 0
-1 - 5a = 0
a = -1/5
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