How to Find Dot Product of 2 Vectors :
Here we are going to see how to find dot product of 2 vectors.
Let a vector and b vector be any two non-zero vectors and θ be the included angle of the vectors. Their scalar product or dot product is denoted by a . b and is defined as a scalar |a vector||b vector| cos θ
Working rule to find scalar product of two vectors
Let
Hence, the scalar product of two vectors is equal to the sum of the products of their corresponding rectangular components.
Question 1 :
Find a vector . b vector when
(i) a vector = i vector−2 vector+k vector and
b = 3i vector − 4j vector−2k vector
Solution :
a_{1} = 1 , a_{2} = -2 and a_{3} = 1
b_{1} = 3 , b_{2} = -4 and b_{3} = -2
a . b = a_{1}b_{1} + a_{2}b_{2 }+ a_{3}b_{3}
a . b = 1(3) + (-2)(-4) + 1(-2)
= 3 + 8 - 2
= 11 - 2
= 9
Hence the dot product of above vectors is 9.
(ii) a = 2i vector + 2j vector − k vector and
b = 6i vector − 3j vector + 2k vector.
Solution :
a_{1} = 2 , a_{2} = 2 and a_{3} = -1
b_{1} = 6 , b_{2} = -3 and b_{3} = 2
a . b = a_{1}b_{1} + a_{2}b_{2 }+ a_{3}b_{3}
a . b = 2(6) + 2(-3) + (-1)2
= 12 - 6 - 2
= 12 - 8
= 4
Hence the dot product of above vectors is 4.
Question 2 :
Find the value λ are perpendicular, where
(i) a = 2i vector + λj vector + k vector and
b = i vector − 2j vector + 3k vector
Solution :
If two vectors are perpendicular a vector . b vector = 0
(2i + λj + k) . (i - 2j + 3k) = 0
2(1) + λ(-2) + 1(3) = 0
2 - 2λ + 3 = 0
5 - 2λ = 0
2λ = 5
λ = 5/2
Hence the value of λ is 5/2.
(ii) a = 2i vector + 4j vector − k vector and
b = 3i vector − 2j vector + λ k vector
Solution :
If two vectors are perpendicular a vector . b vector = 0
(2i + 4j - k) . (3i - 2j + λk) = 0
2(3) + 4(-2) + (-1)(λ) = 0
6 - 8 - λ = 0
-2 - λ = 0
λ = -2
Hence the value of λ is -2.
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