# Define the dot product (scalar product) of two vectors. Which al-gebraic laws are satisfied by dot products? Give examples. When is the dot product of two vectors equal to zero?

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Define the dot product (scalar product) of two vectors. Which al-gebraic laws are satisfied by dot products? Give examples. When is the dot product of two vectors equal to zero?

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Step 1

Let (a1,a2,a3,....) and (b1,b2,b3,...) are two vectors. Then the dot product is defined by:

Step 2

Algebraic laws satisfied by dot product:

(1) (Commutative Property) For any two vectors A and B, A.B = B.A.

(2) (Scalar Multiplication Property) For any two vectors A and B and any real number c, (cA).B = A.(cB) = c(A.B)

(3) (Distributive Property) For any 3 vectors A, B and C, A.(B+C) = A.B + A.C.

Step 3

Examples for:

Commutative Property: (1,2,3).(4,5,6)=1(4)+2(5)+3(6)=4+10+18=32

(4,5,6).(1,2,3)...

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