Question

If `bb(ul u) = << −1,0,2 >>`, `bb(ul v) = << 3,1,2 >>` and `bb(ul w) = << 1,−2,−2 >>` then find?

(a) `3bb(ul v)−2bb(ul u)`

(b) `kbb(ul u) + bb(ul v) + kbb(ul w)`

(c) `|| bb(vec(PQ)) ||` where `bb(vec(OP)) = -3bb(ul u)` and `bb(vec(OQ)) = bb(ul v) + 5bb(ul w)`

(d) Area of the parallelogram with sides `bb(ul u)` and `2bb(ul v)`

Answer 1

We have:

`bb(ul u) = << −1,0,2 >>`
`bb(ul v) = << 3,1,2 >>`
`bb(ul w) = << 1,−2,−2 >>`

Then:

Part (a):

`3bb(ul v)−2bb(ul u) = 3 << 3,1,2 >> - 2<< −1,0,2 >>`
`" " = << 9,3,6 >> - << −2,0,4 >>`
`" " = << 11,3,2 >>`

Part (b):

`kbb(ul u) + bb(ul v) + kbb(ul w) = k<< −1,0,2 >> + << 3,1,2 >> + k<< 1,−2,−2 >>`

`" " = << −k,0,2k >> + << 3,1,2 >> + << k,−2k,−2k >>`
`" " = << −k+3+k,0+1-2k,2k+2-2k >>`
`" " = << 3,1-2k,2 >>`

Part (c):

Let:

`bb(vec(OP)) = -3bb(ul u)`
`" " = -3<< −1,0,2 >>`
`" " = << 3,0,-6 >>`

`bb(vec(OQ)) = bb(ul v) + 5bb(ul w)`
`" " = << 3,1,2 >> + 5<< 1,−2,−2 >>`
`" " = << 3,1,2 >> + << 5,−10,−10 >>`
`" " = << 8,-9,-8 >>`

Then:

`bb(vec(PQ)) = bb(vec(OQ)) - bb(vec(OP))`
`" " = << 8,-9,-8 >> - << 3,0,-6 >>`
`" " = << 5,-9,-2 >>`

Finally:

`|| bb(vec(PQ)) || = || << 5,-9,-2 >> ||`
`" " = sqrt( 5^2 + (-9)^2 +(-2)^2 )`
`" " = sqrt( 25 + 81 + 4 )`
`" " = sqrt( 110 )`

Part (c):

The area is given by the magnitude of the cross product:

`A = || bb(ul u) xx bb(ul v) ||`

we calculate the cross product using:

`bb(ul u) xx bb(ul v) = << −1,0,2 >> xx << 3,1,2 >>`
`\ \ \ = | ( bb(ul(hat i)),bb(ul(hat j)),bb(ul(hat k))), (−1,0,2), (3,1,2) |`

`\ \ \ = | (0,2),(1,2)|bb(ul(hat i)) - | (-1,2),(3,2)|bb(ul(hat j)) + | (-1,0),(3,1)|bb(ul(hat k))`

`\ \ \ = (0-2) bb(ul(hat i)) - (-2-6)bb(ul(hat j)) + (-1-0)bb(ul(hat k))`

`\ \ \ = -2 bb(ul(hat i)) +8bb(ul(hat j)) -bb(ul(hat k))`

Hence,

`A = || bb(ul u) xx bb(ul v) ||`
`\ \ \ = sqrt((-2)^2 + 8^2+(-1)^2)`
`\ \ \ = sqrt(4+64+1) ||`
`\ \ \ = sqrt(69)`