HPC 10.3

10.3 - What You Should Learn

• Find cross products of vectors in space.
• Use geometric properties of cross products of vectors in space.
• Use triple scalar products to find volumes of parallelepipeds.

The Cross Product

• Many applications in physics, engineering, and geometry involve finding a vector in space that is orthogonal to two given vectors.
• The cross product will yield such a vector. It is most conveniently defined and calculated using the standard unit vector form.

EX1: Cross Product Calculation

• Given $u = 6i + 2j + k$ and $v = i + 3j – 2k$, find the following cross products:
  • a. $u \times v$
  • b. $v \times u$
  • c. $v \times v$

EX2: Orthogonality of Cross Product

• Find $u \times v$ and show that it is orthogonal to both u and v.
• $u = <9, 4, 7>$, $v = <6, 3, 2>$

Algebraic Properties

• (Refer to p. 727 for details on algebraic properties).

Geometric Properties

• The cross product is not commutative.
• $u \times v$ and $v \times u$ have equal lengths but opposite directions.
• (Refer to p. 728 for other geometric properties of the cross product of two vectors).

EX3: Finding a Unit Orthogonal Vector

• Find a unit vector that is orthogonal to both u and v.
• $u = <2, 3, 4>$, $v = <0, 1, 1>$

EX4: Area of a Parallelogram

• Find the area of the parallelogram that has the vectors as adjacent sides.

EX5: Area of a Triangle

• Find the area of the triangle with the given vertices: (1, –4, 3), (2, 0, 2), (–2, 2, 0).

Triple Scalar Product

• For vectors u, v, and w in space, the dot product of u and $v \times w$ is called the triple scalar product of u, v, and w.

Volume of a Parallelepiped

• When the vectors u, v, and w do not lie in the same plane, the triple scalar product $u \cdot (v \times w)$ can be used to determine the volume of the parallelepiped (a polyhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges.

EX6: Volume of a Parallelepiped

• Find the volume of the parallelepiped having adjacent edges u, v, and w.