Question

Find the projection of `bar(AB)` on `bar(CD)`, where A ≡ (2, –3, 0), B ≡ (1, –4, –2), C ≡ (4, 6, 8), D ≡ (7, 0, 10).

Answer 1

Let `bara, barb, barc` and `bard` be the position vectors of A, B, C and D respectively with respect to the origin O.

∴ `bara = 2hati - 3hatj`,

`barb = hati - 4hatj - 2hatk`,

`barc = 4hati + 6hatj + 8hatk`,

`bard = 7hati + 10hatk`

∴ `bar(AB) = barb - bara`

= `(hati - 4hatj - 2hatk) - (2hati - 3hatj)`

= `-hati - hatj - 2hatk`

and `bar(CD) = bard - barc`

= `(7hati + 10hatk) - (4hati + 6hatj + 8hatk)`

= `3hati - 6hatj + 2hatk`

∴ `bar(AB)*bar(CD) = (-hati - hatj - 2hatk)*(3hati - 6hatj + 2hatk)`

= (–1)(3) + (–1)(–6) + (–2)(2)

= – 3 + 6 – 4

= – 1

`|bar(CD)| = sqrt(3^2 + (-6)^2 + 2^2)`

= `sqrt(9 + 36 + 4)`

= 49

= 7 

∴ Projection of `bar(AB)` on `bar(CD)`

= `(bar(AB)*bar(CD))/|bar(CD)|`

= `-1/7`