The angle between the two vectors `vecA=3hati+4hatj+5hatk` and `vecB=3hati+4hatj-5hatk` will be :

Answer

(i) `hat(i)+that(j)+hat(k)`, (ii) `sqrt(t^(2)+2)`, (iii) `hat(j)`, (iv) `t//sqrt(t^(2)+2)`, (vi) `sqrt(2)//sqrt(t^(2)+2)`

Explanation

`vec(r)=that(i)+t^(2)/2hat(j)+that(k)`
(i) `vec(v)=(dvec(r))/(dt)=hat(i)+that(j)+hat(k)" "` (iii) speed `|vec(v)|=sqrt(t^(2)+2)`
(iii) `vec(a)=(dvec(v))/(dt)=hat(j)" "` (iv) `|vec(a)|=1`
(v) `vec(a)_(T)=(vec(a).hat(v))hat(v)=([hat(j)((hat(i)+that(j)+hat(k)))/(sqrt(t^(2))+2)])((hat(i)+that(j)+hat(k)))/sqrt(t^(2)+2)`
`vec(a)_(T)=(t/sqrt(t^(2)+2))hat(v)=(t(hat(i)+that(j)+hat(k)))/((t^(2)+2)), |vec(a)_(T)|=t/sqrt(t^(2)+2)`
As `a_(N)^(2)+a_(T)^(2)=a^(2)`
so `a_(N)=sqrt(a^(2)-a_(T)^(2))=sqrt(2)/sqrt(t^(2)+2)`