Question

Find a vector of magnitude 21 units in the direction opposite to that of `vec(AB)` where A and B are the points A(2, 1, 3) and B(8, –1, 0) respectively.

Answer 1

Given: `vecA = 2hati + hatj + 3hatk`

`vecB =8hati - hatj + 0hatk`

`vec(AB) = vec(B) - vec(A)`

= `(8hati - hatj) - (2hati + hatj + 3hatk)`

= `6hati + 2hatj + 3hatk`

In the opposite direction

`vec(BA) = -(6hati - 2hatj - 3hatk)`

= `-6hati + 2hatj + 3hatk`

a unit vector of magnitude 21 is

`vecu ("suppose") = (λ(vec(BA)))/(|vec(BA|)`

= `(21(-6hati + 2hatj + 3hatk))/(|-6hati + 2hatj + 3 hatk|)`

= `(21(-6hati + 2hatj + 3hatk))/(sqrt(36 + 4 + 9))`

= `21/7(-6hati + 2hatj + 3hatk)`

= `3(-6hati + 2hatj + 3hatk)`

∴ ` vecu = -18 hati + 6hatj + 9hatk`