SECTION 7.3 The Scalar Product of Two Vectors5. For any two vectors A and B, show that AB AB+ AB+SAB. Suggestions: Write A and B in unit-vector form and useEquations 7.4 and 7.5.

Dot Product of two vectors The dot product is also known as the scalar product. The dot product of two vectors A→ and B→ is given by the relation A→.B→=AB cosθ θ is the angle between the vectors A→ and B→ and A and B are the respective magnitudes. The unit vectors in the direction of x, y and z axis are i^, j^ and k^ respectively. They are mutually orthogonal to each other. This means that the dot product between any two of them is equal to zero as the angle between them is 90°. Thus i^.j^=0i^.k^=0j^.k^=0 If the dot product is performed with itself, the answer will be unity as the angle the unit vector makes it itself is 0° i^.i^=1j^.j^=1k^.k^=1Let there be two vectors A→ and B→ be written in terms of rectangular components. A→=Axi^+Ayj^+Azk^B→=Bxi^+Byj^+Bzk^ Then the dot product between these two vectors will be A→.B→=Axi^+Ayj^+Azk^.Bxi^+Byj^+Bzk^=AxBxi^.i^+AxByi^.j^+AxBzi^.k^+AyBxj^.i^+AyByj^.j^+AyBzj^.k^+AzBxk^.i^+AzByk^.j^+AzBzk^.k^ From the previous step, we have seen i^.j^=i^.k^=j^.i^=j^.k^=k^.i^=k^.j^=0i^.i^=j^.j^=k^.k^=1 Substituting this in the above expression we get A→.B→=AxBx+AyBy+AzBzAnswer: We have proved, if two vectors A→ and B→ are expressed in terms of their rectangular components then A→.B→=AxBx+AyBy+AzBy

Answered: - For any two vectors A and B, show…

- For any two vectors A and B, show that A B AB+ AB+ AB. Suggestions: Write A and B in unit-vector form and use Equations 7.4 and 7.5.