The Cross Product of 2 Vectors 11.3 JMerrill, 2010

Unit Vectors in 2D In 2-D space, the unit vectors and are the standard unit vectors and denoted by i = and j = j = i =

Unit Vectors in 2D Any vector can be written as a linear combination of the vectors I and j. v = <v 1, v 2 > = v 1 <1,0> + v 2 <0,1> = v 1 i + v 2 j The scalars v 1 and v 2 are the horizontal and vertical components of v.

Writing a Linear Combination of Unit Vectors u has initial point (2, -5) and terminal point (-1,3). Write u as a linear combination of the unit vectors i & j. u = <-1-2, 3-(-5)> = <-3, 8> = -3i + 8j

Unit Vectors in 3D

The Cross Product

Finding The Cross Product An easy way to calculate the cross product is to use a matrix. We use the determinant form with cofactor expansion.

Finding The Cross Product An easy way to calculate the cross product is to use a matrix. We use the determinant form with cofactor expansion.

Finding the Cross Product Subtraction sign Addition Sign

Example Given u = i + 2j + k and v = 3i + j + 2k, find the cross product of u x v.

You Try Given u = i + 2j + k and v = 3i + j + 2k, find the cross product of v x u.

Using the Cross Product Find a unit vector that is orthogonal to both u = 3i – 4j + k and v = -3i + 6j. The cross product gives a vector that is orthogonal to both u and v = -6i – 3j + 6k The question asks for a unit vector that’s orthogonal.

So, we need to divide by the magnitude of the orthogonal vector. -6i – 3j + 6k

Triple Scalar Product Given 3 vectors u = 3i – 5j + k v = 2j – 2k w = 3i + j + k Find the volume of a parallelepiped having these vectors as adjacent edges. The volume is found by V = |u∙(v x w)|

Triple Scalar Product