The crossproduct of the dots are

Question

You are looking down at a map. A vector U with |U| 1 points north and a vector V with |V|= 9 points northeast. The crossproduct UxV points:

A) south?
B) northwest?
C) up?
D) down ?

 

The magnitude |UxV|=?

Answer 1

Dalia,

 

Define an orthogonal coordinate system with unit vectors: (East, North, Up). Then in this system:

vector u = (0,1,0)

vector v = (9 cos [45^o], 9 sin[45^o], 0)  since northeast is midway between East and North so angle = (1/2)90^o =45^o.

But sin[45o] = cos[45o] = (1/2)√2 = 0.7071, so

vector v = (6.3639, 6.3639,0)

 

Using the Right Hand Screw (RHS) Rule: when your fingers of the right hand point in the direction of the first vector (u) and sweep it toward the second vector (v), your thumb points to the direction of the cross product vector.  So in this case North sweeps Eastward and the thumb points DOWN given you're looking down on the map.  Answer D.

 

So u X v = (0,1,0) X (6.3639, 6.3639,0) = [(0)(6.3639) - 1(6.3639) + (0)(0)] unit vector k =  - 6.3639 unit vector k

 

Or another way to look at a cross product is a vector whose direction is given by the RHS rule, and whose magnitude is the area of the parallelogram between the two vectors = |u| |v| sin Θ, where that's the included angle.  Remember, that |u| sin Θ is the perpendicular height of the parallelogram and |v| is its base, so the product is the area.

 

Also, if you flip the vectors and look at vector v X vector u then its direction gets flipped to UP by the RHS rule. The magnitude or area stays the same.