Find the angles at which the following vectors are inclined to each of the coordinate axes :

i.


ii.


iii.

(i)

Let, r⃗ = î - j + k̂


The direction ratios of the vector r⃗ = 1, - 1, 1


And, |r⃗ | = √ ((1)2 + ( - 1)2 + (1)2)


= √3


The direction cosines of the vector r⃗



So,








Thus, angles made by with the coordinate axes are given by


(ii)


Let, r⃗ = 0 î + j - k̂


The direction ratios of the vector r⃗ = 0, 1, - 1


And, |r⃗ | = √ ((0)2 + (1)2 + ( - 1)2)


= √2


The direction cosines of the vector r⃗



So,












Thus, angles made by with the coordinate axes are given by


(iii)


Let, r⃗ = 4î + 8 j + k̂


The direction ratios of the vector r⃗ = 4, 8, 1


And, |r⃗ | = √ ((4)2 + (8)2 + (1)2)


= √81


= 9


The direction cosines of the vector r⃗



So,








Thus, angles made by with the coordinate axes are given by


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