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