Calculating the Scalar Product Using Components
We can calculate the scalar product Ä • B directly if we know the x-, y-, and 2-components of A and B. To see how this is done, let’s first work out the scalar products of the unit vectors. This is easy, since i, j, and k all have magnitude 1 and are perpendicular to each other. Using dot product, we find
i•i = j•j= k•k = (1)(1) cos 0° = 1
i•j= i•k = j•k = (1)(1) cos 0° = 1
Now we express Ä and B in terms of their components, expand the product, and use these products of unit vectors:
A – B = (Axi+ Ayj + Azk) • (Bxi + Byj + Bzk)
= (Axi • Bxi + Axi • Byj + Axi • Bzk)
+ (Ayj • Bxj + Ayj • Byj + Ayj • Bzk)
+(Azk • Bxi + Azk • Byj + Azk • Bzk)
The above equation will now give us:
AxBx(i • I) + AxBy(i • j) + AxBz(i • k)
+ AyBx(j•i) + AyBy(j•j) + AyBz(j•k)
+ AzBx(k•i) + AzBy(k•j) + AzBz (k•k)
We can now see that six of these nine terms are zero, and the three that survive give simply: A•B = AxBx + AyBy + AzBz [which is a scalar (dot) product in terms of components]. Thus the scalar product of two vectors is the sum of the products of their respective components. The scalar product gives a straightforward way to find the angle between any two vectors A and B whose components are known.