Let ` vecf(t)=[t] hat i+(t-[t]) hat j+[t+1] hat k , w h e r e[dot]` denotes the greatest integer function. Then the vectors ` vecf(5/4)a n df(t),0lttlt1` are(a) parallel to each other(b) perpendicular(c) inclined at `cos^(-1)2 (sqrt(7(1-t^2)))` (d)inclined at `cos^(-1)((8+t)/sqrt (1+t^2))`;

Options

A. parallel to each other
B. perpendicular to each other
C. inclined at `cos^(-1) 2/(sqrt7(1-t^(2))`
D. inclined at `cos ^(-1) (8+t)/(9sqrt(1+t^(2)))`

Answer

d

Explanation

`vecf""(5/4)= [5/4] hati+ (5/4-[5/4])hatj + [5/4=1]hatk`
`hati+ (5/4-1)hatj + 2hatk`
`hati + 1/4 hatj + 2hatk`
when `0 lt t 1 ,vecf "" (t) = 0 hati + { t-0} hatj +hatk`
` hatj +hatk`
` vecf "" (5/4) .vecf"" (t) = 2+t/4`
so, `costheta= 4/(|vect+1/4hatj + 2hatk||thatj +hatk|)`
`(2+t/4)/(sqrt(1+1/16=4)sqrt(1+t^(2)))`
` ( 8 +t)/(9sqrt(1 + t^(2)))`