10.7: Direction Angles and Direction Cosines

Direction Angles and Direction Cosines

OBJECTIVE: given a vector, find its direction angles and direction cosines and vice versa

The graph shows a vector v and its direction angles:




the first three letters of the Greek alphabet alpha, beta, and, gamma are used for the three direction angles. like x, y, and z, the letters come in alphabetical order, corresponding to the three axes.

DEFINITIONS: Direction angles and Direction cosines:

the direction angles of a position vector are:

alpha, from the x-axis to the vector

beta, from the y-axis to the vector

gamma, from the z-axis to the vector

the direction cosines of a position vector are the cosines of the direction angles:

c1 = cos (alpha)

c2 = cos (beta)

c3 = cos (gamma)

PROPERTIES: Direction Cosines:

Pythagorean Property of Direction Cosines:

if alpha, beta, and gamma are the direction angles of a position vector and c1 = cos (alpha), c2 = cos (beta), and c3 = cos (gamma) are the direction cosines, then

cos squared (alpha) + cos squared (beta) + cos squared (gamma) = 1

or

c1 squared + c2 squared + c3 squared = 1

Unit Vector Property of Direction cosines:

vector u = c1 vector i + c2 vector j + c3 vector k is a unit vector in the direction of the given vector.

Example Problem:

find the direction cosines and the direction angles for vector v = 3 vector i + 7 vector j + 5 vector k

Solution:

find the dot products of vector v dot vector i, vector v dot vector j, vector v dot vector k. then use these to find the angles.

vector v dot vector i = (3 vector i + 7 vector j + 5 vector k) (1 vector i + 0 vector j + 0 vector k) = 3, which is the coefficient of vector i in vector v

vector v dot vector i = (3 vector i + 7 vector j + 5 vector k) (0 vector i + 1 vector j + 0 vector k) = 7, which is the coefficient of vector j in vector v

vector v dot vector i = (3 vector i + 7 vector j + 5 vector k) (0 vector i + 0 vector j + 1 vector k) = 5, which is the coefficient of vector k in vector v

absolute value of vector v = square root of 3 squared + 7 sqaured + 5 squared = squared root of 83

abolute value of vector i = abolute value of vector j = abolute value of vector k = 1
vectors i, j, and k are unit vectors

square root of 83(1) cos alpha = 3 ==> cos alpha = 3/square root of 83 ==> alpha = 70.774 degrees

square root of 83(1) cos beta = 7 ==> cos beta = 7/square root of 83 ==> beta = 39.794 degrees

square root of 83(1) cos gamma = 5 ==> cos gamma = 5/square root of 83 ==> gamma = 56.713 degrees

for addtional help, go here: http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node52.html

Kaori, you're up next! Good luck!

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