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The acute angle between the line r=(hat(...

The acute angle between the line `r=(hat(i)+2hat(j)+hat(k))+lamda(hat(i)+hat(j)+hat(k))` and the plane `r*(2hat(i)+hat(j)+hat(k))=5`

A

`cos^(-1)((sqrt(2))/(3))`

B

`sin^(-1)((sqrt(2))/(3))`

C

`tan^(-1)((sqrt(2))/(3))`

D

`sin^(-1)((sqrt(2))/(sqrt(3)))`

Text Solution

Verified by Experts

The correct Answer is:
B
We know that, the angle between the line and the plane `r*n=d` is
`sintheta=(n*b)/(|n||b|)`
Given equation of line is `r=(hat(i)+2hat(j)+hat(k))+lamda(hat(i)+hat(j)+hat(k))and` planeis `r*(2hat(i)-hat(j)+hat(k))=5`
Here, `b=hat(i)+hat(j)+hat(k)andn=2hat(i)-hat(j)+hat(k)`
`:." "sintheta=((2hat(i)-hat(j)+hat(k))*(hat(i)+hat(j)+hat(k)))/(|2hat(i)-hat(j)+hat(k)|hat(i)+hat(j)+hat(k)|)`
`=(2-1+1)/(sqrt(2^(2)+(-1)^(2)+(1)^(2))sqrt(1^(2)+1^(2)+1^(2)))`
`=(2)/(sqrt(4+1+1)sqrt(1+1+1))=(2)/(sqrt(6)sqrt(3))=(2)/(3sqrt(2))`
`rArr" "sintheta=(sqrt(2))/(3)`
`rArr" "theta=sin^(-1)((sqrt(2))/(3))`
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