The direction angles of a vector v are the angles between the vector and the coordinate axes (x-axis, y-axis, z-axis). If α, β, and γ represent the direction angles, the vector’s direction cosines are given by:
cosα,cosβ,cosγ
The sum of the squares of the direction cosines must always be equal to 1:
cos2α+cos2β+cos2γ=1
This relationship is derived from the Pythagorean theorem and is a key property of direction cosines.
Solution
(i) Check if α=45∘, β=45∘, γ=60∘ can be direction angles
To verify if these angles can be direction angles, we calculate:
cos2α+cos2β+cos2γ
Find the cosine values:
cos45∘=21,cos60∘=21
Substitute the values:
cos245∘+cos245∘+cos260∘=(21)2+(21)2+(21)2
Simplify:
=21+21+41=42+1=45
Since the result is 45=1, the given angles cannot be direction angles.
(ii) Check if α=30∘, β=45∘, γ=60∘ can be direction angles
Again, calculate:
cos2α+cos2β+cos2γ
Find the cosine values:
cos30∘=23,cos45∘=21,cos60∘=21
Substitute the values:
cos230∘+cos245∘+cos260∘=(23)2+(21)2+(21)2
Simplify:
=43+21+41=43+2+1=46=23
Since the result is 23=1, the given angles cannot be direction angles.
(iii) Check if α=45∘, β=60∘, γ=60∘ can be direction angles
Now, calculate:
cos2α+cos2β+cos2γ
Find the cosine values:
cos45∘=21,cos60∘=21
Substitute the values:
cos245∘+cos260∘+cos260∘=(21)2+(21)2+(21)2
Simplify:
=21+41+41=42+1+1=44=1
Since the result is 1, the given angles can be direction angles.
Key Formulas or Methods Used
Direction Cosines: The cosines of the angles between a vector and the coordinate axes are called the direction cosines.
Condition for Direction Angles: The direction angles α, β, and γ of a vector must satisfy:
cos2α+cos2β+cos2γ=1
Summary of Steps
(i) Calculate cos2α+cos2β+cos2γ for α=45∘, β=45∘, γ=60∘ and find that it does not equal 1, so they cannot be direction angles.
(ii) Calculate cos2α+cos2β+cos2γ for α=30∘, β=45∘, γ=60∘ and find that it does not equal 1, so they cannot be direction angles.
(iii) Calculate cos2α+cos2β+cos2γ for α=45∘, β=60∘, γ=60∘ and find that it equals 1, so they can be direction angles.