666 CHAPTER 9 Polar Coordinates; Vectors Based on equation (6), when two direction cosines are known, the third is determined up to its sign. Knowing two direction cosines is not sufficient to uniquely determine the direction of a vector in space. Finding a Direction Angle of a Vector The vector v makes an angle of 3 α π = with the positive x -axis, an angle of 3 β π = with the positive y -axis, and an acute angle γ with the positive z -axis. Find .γ Solution EXAMPLE 9 By equation (6), we have π π γ γ γ γ γ γ π γ π ( ) ( ) ( ) ( ) + + = + + = = = = − = = cos 3 cos 3 cos 1 1 2 1 2 cos 1 cos 1 2 cos 2 2 or cos 2 2 4 or 3 4 2 2 2 2 2 2 2 0 2 γ π < < Since γ must be acute, 4 . γ π = The direction cosines of a vector give information about only the direction of the vector; they provide no information about its magnitude. For example, any vector that is parallel to the xy -plane and makes an angle of 4 π radian with the positive x -axis and y -axis has direction cosines α β γ = = = cos 2 2 cos 2 2 cos 0 However, if the direction angles and the magnitude of a vector are known, the vector is uniquely determined. Writing a Vector in Terms of Its Magnitude and Direction Cosines Show that any nonzero vector v in space can be written in terms of its magnitude and direction cosines as EXAMPLE 10 Solution Let a b c v i j k. = + + From the equations in (5), note that α β γ = = = a b c v v v cos cos cos Substituting gives α β γ α β γ [ ] ( ) ( ) ( ) ( ) ( ) ( ) = + + = + + = + + a b c v i j k v i v j v k v i j k cos cos cos cos cos cos α β γ [ ] ( ) ( ) ( ) = + + v v i j k cos cos cos (7) Now Work PROBLEM 59 Example 10 shows that the direction cosines of a vector v are also the components of the unit vector in the direction of v .
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