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12.4 THE CROSS PRODUCT / Vector Product Illustration of cross product

Definition: Cross Product

u x v = (( u(( u( sin( ) n

where n is a unit vector.

Parallel Vectors

Nonzero vectors u and v are parallel if and only if u x v = 0Properties of the Cross Product

If u, v and w are any vectors and r, s are scalars, then

1. (r u)x (s v) = (rs)(u x v)

2. u x (v + w) = (u x v) + (u x w)

3. (v + w) x u = (v x u) + (w x u)

4. v x u = - (u x v)

5. 0 x u = 0Area of a Parallelogram - ( u x v( ( u x v( = ( u(( v(( sin (( n = ( u(( v(sin ( Determinant Formula for u x v

Calculating Cross Product Using DeterminantIf u = u1i + u2j + u3k and v = v1i + v2j + v3k, then u x v =

Example:1

Find u X v and v X u if u = 7i + j k and v = -2i + 4j + kEquation of the plane:

Example:2

Find the following:

(a) a vector perpendicular to the plane of P (1,2,3), Q (-5,4,1) and R ( 3,-1,2).

(b) a unit vector perpendicular to the plane of P (1,2,3), Q (- 5,4,1) and R ( 3,-1,2).

Example:3

Find the area of the triangle with vertices P (1,2,3), Q (-3,4,5) and R ( 6,-1,2).

Triple Scalar or Box Product

Absolute magnitude

Value

Calculating the Triple Scalar Product

Example:5

Find the volume of an oblique box (parallelepiped) determined by the vertices P (-3,1,6) , Q (-4,3,1) , R (5,-2,3) and S (3,2,1).

(Optional)Describe about Torque Vector

EMBED Equation.3

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