mcv4u1 5.4 - the cross product of two vectors the cross product also called a "vector...

MCV4U1

5.4 - The Cross Product Of Two Vectors

The cross product also called a "vector product" only exists in R3. a CROSS b, produces a vector quantity that is perpendicular to BOTH a and b.

If a = (x1, y1, z1) and b = (x2, y2, z2)

Cross Product Formula:

=(y1z2 - y2z1, z1x2 - z2x1, x1y2 - x2y1)

a

ba x b

Cross Product Shortcut!!!

Instead of memorizing the formula for the cross product try usingthe following shortcut.

1.) Eliminate (ignore) the component column that you are trying to calculate.

2.) Calculate: Down Product - Up Product.

= x1 y1 z1 x1

x2 y2 z2 x2

=(y1z2 - y2z1, z1x2 - z2x1, x1y2 - x2y1)

Ex.) Calculate the cross product of the following pairs of vectors.

a) a = (6, -1, 3) and b = (-2, 5, 4)

b) u = (4, -6, 7) and v = (1, 3, 2)

Magnitude of the cross product

* Where θ is the angle between the vectors

Ex.) If and the angle between them is 30 find

Ex.) If a = (3, -1, -5) and b = ( 7, -3, 0) find:

a)

b)

c) A unit vector perpendicular to BOTH a and b.

Properties of the Cross Product

1) Commutative Law is NOT true.

2) Distributive Property

3) Scalar Multiplication

4)

However

AND

Homework: p.185 # 1 - 7, 9, 15

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