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प्रश्न
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).
उत्तर
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`
