Given `underset~a=(3,-2,1)` and `underset~b=(4,3,-4)`, verify numerically that
`underset~a*underset~b=abs(underset~a)abs(underset~b)cos theta=x_1x_2+y_1y_2+z_1z_2`
where `underset~a=(x_1,y_1,z_1)` and `underset~b=(x_2,y_2,z_2)` (4 marks)
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`text{Scalar Product result 1:}`
| `underset~a*underset~b` | `=x_1x_2+y_1y_2+z_1z_2` | |
| `=3xx4+(-2)xx3+1xx(-4)` | ||
| `=2` |
`text{Scalar Product result 2:}`
`underset~a*underset~b=abs(underset~a)abs(underset~b)cos theta`
`text{Let}\ \ underset~c=underset~b-underset~a`
`underset~c=((4),(3),(-4))-((3),(-2),(1))=((1),(5),(-5))`
`abs(underset~c)=sqrt(1^2+5^2+(-5)^2)=sqrt(51)`
`abs(underset~a)=sqrt(3^2+(-2)^2+1^2)=sqrt(14)`
`abs(underset~b)=sqrt(4^2+3^2+(-4)^2)=sqrt(41)`
`text{Using cosine rule:}\ \ c^2=a^2+b^2-2ab\ cosC`
`=>\ \ ab\ cosC=(a^2+b^2-c^2)/2`
| `underset~a*underset~b` | `=abs(underset~a)abs(underset~b)cos theta` | |
| `=(abs(underset~a)^2+abs(underset~b)^2-abs(underset~c)^2)/2` | ||
| `=(14+41-51)/2` | ||
| `=2` |
`:.underset~a*underset~b=abs(underset~a)abs(underset~b)cos theta=x_1x_2+y_1y_2+z_1z_2`