1. Multiplication of vector by a scalar
Let vector a is multiplied by a scalar m. If m is a positive quantity, only magnitude of the vector will change by a factor ‘m’ and its direction will remain same. If m is a negative quantity the direction of the vector will be reversed.
2. Multiplication of a vector by a vector
(i) Dot product or scalar product
(ii) Cross product or vector product
Dot product or scalar product
The dot product of two vectors a and b is defined as
a-> . b-> = ab cosθ
where a and b are the magnitudes of the respective vectors and θ is the angle between them. The final product is a scalar quantity. If two vectors are mutually perpendicular then θ = 900 and cos 90 = 0, Hence, their dot product is zero.
Some examples of dot product: work = F-> . s-> = Fs cosθ
Here,
The dot product obeys commutative law
i.e. a ->. b-> =b-> .a->
Hence, a-> . b-> = axbx + ayby + azbz
Illustration :
Find the angle between the vectors A and B where
Solution :
We know
A->.B-> = |A||B| cosθ where |A| = √(22 +32 + 32) = √22,
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