Calculus Question

Question

Please answer the following questions and show all steps. Thank you in advance.

1. Given the points X(0, 2, –1), Y(3, 1, 2), and Z(–3, 2, 5), find (Vector XY × Vector XZ) • Vector YZ.
2. If vector b = [3, 6, 2] and vector c = [0, 0, 8], find a unit vector perpendicular to both vector b and c.
3. An airplane is on a heading of 075° at 250 km/h and an 85 km/h wind is blowing from 300°. Determine the resultant velocity of the airplane, using Cartesian vectors, and state the groundspeed and bearing of the resultant.
4. Given the points A(1, 2, 3), B(5, 4, 3), and C(2, 1, 2), find ∠ABC

Answer 1

Hey Jo!

I'll chime in and get you started with number 1.

1. Let's start by calculating the vectors XY, XZ, and YZ

XY = <3,-1,3>

XZ = <-3,0,6>

YZ = <-6,1,3>

Having these vectors calculated, let's computer the expression you listed by starting with the cross product. I'll do my best to make the determinant form of the cross product clear.

i j k

3 -1 3

-3 0 6

Carrying out the determinant calculation we have

(-6)i

(-27)j

(-3)k

Now we can take the dot product with vector YZ.

<-6,-27,-3>•<-6,1,3> = 36 - 27 - 9 = 0

What does this tell us exactly? If the dot product of two non-zero vectors is zero, then they are orthogonal to each other. The cross product of XY and XZ gave us a vector orthogonal to the plane spanned by the two of them, and now we find that vector is orthogonal to YZ. This implies that YZ also lies in the plane spanned by XY and XZ!