Find the cross product of a x b and prove that it is orthogonal to a and b: a =

Question

Find the cross product of a x b and prove that it is orthogonal to a and b: a = ti + cos(t)j + sin(t)k and b = i - sin(t)j + cos(t)k (where i, j, and k are unit vectors). [I mostly need help with proving that the cross product is perpendicular to a and b] Find the cross product of a x b and prove that it is orthogonal to a and b: a = ti + cos(t)j + sin(t)k and b = i - sin(t)j + cos(t)k (where i, j, and k are unit vectors). [I mostly need help with proving that the cross product is perpendicular to a and b]

Explanation / Answer

Solution:

a x b =
|i.......j...........k|
|t...cos t....sin t|

= <cos2(t) + sin2(t), -(t cos t - sin t), -t sin t - cos t>

|1 - sint cost

So, a x b = <1, -t cos t + sin t, -t sin t - cos t>.

Now to verify orthogonality with a and b, compute the respective dot products and show they each equal 0

(a x b) · a = <1, -t cos t + sin t, -t sin t - cos t> · <t, cos t, sin t>

               = t + (-t cos2(t) + sin t cos t) + (-t sin2(t) - sin t cos t)

               = t - t (cos2(t) + sin2(t))

               = t - t

               = 0

(a x b) · b = <1, -t cos t + sin t, -t sin t - cos t> · <1, -sin t, cos t>

               = 1 + (t sin t cos t - sin2(t)) + (-t sin t cos t - cos2(t))

               = 1 - (sin2(t) + cos2(t))

               = 1 - 1

               = 0

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