1. Check whether the following vectors are perpendicular to each other.
(4i + 2j - 8k) and (3i - 2j + k)
2. Check whether the following vectors are perpendicular to each other.
(i - 5j + 7k) and (-2i + 3j +4k)
3. Show that the vectors a = 2i + 3j + 6k, b = 6i + 2j − 3k, and c = 3i − 6j + 2k are mutually orthogonal.
4. 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.
5. If the vectors (i + 2j - 5k) and (3i - 2j + ak) are perpendicular to each other, find the value of 'a'.
1. Answer :
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 to each other.
2. Answer :
Find the dot product of the given two vectors :
= (i - 5j + 7k) ⋅ (-2i + 3j + 4k)
= 1(-2) - 5(3) + 7(4)
= -2 - 15 + 28
= 11 ≠ 0
Since the dot product is not equal to zero, the given two vectors are not perpendicular to each other.
3. Answer :
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.
4. Answer :
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.
5. Answer :
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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