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Published byCole Rollins Modified over 5 years ago

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11.4 The Cross product For an animation of this topic visit: Red x Green = Blue

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**What is the cross product?**

a × b is a vector that is perpendicular to both a and b. ||a × b|| is the area of the parallelogram spanned by a and b (i.e. the parallelogram whose adjacent sides are the vectors a and b). The direction of a×b is determined by the right-hand rule. (This means that if we curl the fingers of the right hand from a to b, then the thumb points in the direction of a × b.)

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**Definition of Cross Product of Two Vectors in Space**

Find the cross product of vectors (by hand and with a calculator) u = 3i +2j – k v = i + 2k Note: To find a cross product on the TI89 Press 2nd 5 (math) – 4 matrix – L Vector ops - crossP crossP([3,2,-1],[1,0,2])

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**Definition of Cross Product of Two Vectors in Space**

For an explanation and animation of the cross product visit:

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**Example 1 Find the cross product of the two vectors**

u = i – 2j + k v = 3i + j – 2k Find u x v Find v x u Find v x v

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**Example 1 Find the cross product of the two vectors**

u = i – 2j + k v = 3i + j – 2k Find u x v Find v x u Find v x v

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**Example 1 continued u = i – 2j + k v = 3i + j – 2k**

Find a unit vector that is orthogonal to u and v

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**The Triple Scalar Product**

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**Geometric Property of Triple Scalar**

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**Example 5 u = 3i – 5j + k v = 2j – 2k w = 3i + j + k**

Find the volume of the parallelepiped having vectors u, v and w for sides

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Example 5 hint This works because bxc yields a vector perpendicular to a and b in with the magnitude of the area of the parallelogram formed (the base of the parallel piped) bxc points in the direction of the height of the parallelepiped when dotted with a this gives the magnitude of bxc times the portion of a that points in the direction of bxc (the direction of the height )in other words this gives us the same as the formula Bh

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**Q: Why should you never make a math teacher angry?**

A: You might get a cross product Q: What do you get when you cross an elephant and a banana? A: | elephant | * | banana | * sin(theta)

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Proof of the cross product proof that the cross product is orthogonal to the two original vectors is part of the homework │u x v │ = │u │ │v │sinθ

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(a) Define vector product (b) Understand the properties of vector product (c)Find the area of parallelogram.

(a) Define vector product (b) Understand the properties of vector product (c)Find the area of parallelogram.

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