Let `vec(a)=2hat(i)+3hat(j)-hat(k) , vec(b)=-hat(i)+3hat(j)+4hat(K)`. Evaluate
(i)`|vec(a)|,|vec(b)|`
(ii) `vec(a).vec(b)`
(iii) the angle between the vectors `vec(a) and vec(b)`
(iv) the projection of `vec(a) on vec(b)`
(v) the projection of `vec(b) on vec(a)`
area of the `DeltaAOB` where O is origin

Explanation

Give `vec(a)=2hat(i)+3hat(j)-hat(k),vec(b)=-hat(i)+3hat(j)+4hat(k)`
(i) `|vec(a)|=sqrt(2^(2)+3^(2)+(-1)^(2))=sqrt(4+9+1)=sqrt(14)`
`|vec(b)|=sqrt((-1)^(2)+3^(2)+4^(2))=sqrt(1+9+16)= sqrt(26)`
(ii)`vec(a).vec(b)=2(-1)+3xx3+(-1)(4)=3`
(iii) The angle `theta` between the vectors `vec(a)` and `vec(b)` is given by
`cos theta =(vec(a).vec(b))/(|vec(a)||vec(b)|)=(3)/(sqrt(14)sqrt(26))=(3)/(2sqrt(91))`
(iv) the projection of `vec(a)` on `vec(b) =|vec(a)| cos theta`
`sqrt(14)xx(3)/(sqrt(14)sqrt(26))=(3)/(sqrt(26))`
(v) The projection of `vec(b)` on `vec(a)=|vec(b)|cos theta`
`sqrt(26)xx(3)/(sqrt(14)sqrt(26))=(3)/(sqrt(14))`
(vi) Area of `DeltaAOB=(1)/(2)|vec(a)||vec(b)| sin theta`
Now `sin ^(2) theta=1-cos^(2) theta=1-((3)/(2sqrt(91)))^(2)`
`=1-(9)/364=(355)/(364)`
Area of `DeltaAOB=(1)/(2)sqrt(14)sqrt(26).sqrt((355)/364)`
`=sqrt((355)/(2))=9.42 sq`. unit approx.