Question

A bird flies through a distance in a straight line given by the vector `hati + 2hatj + hatk`. A man standing beside a straight metro rail track given by `vecr = (3 + λ)hati + (2λ − 1)hatj + 3λhatk` is observing the bird. The projected length of its flight on the metro track is ______.

Options

• `6/sqrt(14)` units

• `14/sqrt(6)` units

• `8/sqrt(14)` units

• `5/sqrt(6)` units

Answer 1

A bird flies through a distance in a straight line given by the vector `hati + 2hatj + hatk`. A man standing beside a straight metro rail track given by `vecr = (3 + λ)hati + (2λ − 1)hatj + 3λhatk` is observing the bird. The projected length of its flight on the metro track is `underlinebb(8/sqrt(14) units)`.

Explanation:

Given, bird vector = `hati + 2hatj + hatk`

And, man standing on line:

`vecr = (3 + λ)hati + (2λ − 1)hatj + 3λhatk`

`vecr = (3hati - hatj) + λ(hati + 2hatj + 3hatk)`

We need to find the projection of the bird vector on the line.

Thus, we need to find the projection of the bird vector `(hati + 2hatj + hatk)` on the parallel vector of the line `(hati + 2hatj + 3hatk)`.

Here, `veca = hati + 2hatj + hatk`

`vecb = hati + 2hatj + 3hatk`

Now, projection of `veca` along `vecb` = `(veca*vecb)/|vecb|`

= `((hati + 2hatj + hatk) * (hati + 2hatj + 3hatk))/(|hati + 2hatj + 3hatk|)`

= `((hati + 2hatj + hatk) * (hati + 2hatj + 3hatk))/(sqrt(1^2 + 2^2 + 3^3))`

= `(1 xx 1 + 2 xx 2 + 1 xx 3)/(sqrt(1^2 + 2^2 + 3^3))`

= `(1 + 4 + 3)/(sqrt(1 + 4 + 9))`

= `8/sqrt14` units