Question

If a unit vector $\vec a$ makes angles with $\frac {\pi}{3} \ with \hat i,\frac {\pi}{4}\ with\ \hat j$ and an acute angle θ with $\hat k$, then find θ and hence, the components of $\vec a$ .

Answer 1

Angle which the vector a makes with the x axis = π/3

Angle which the vector a makes with the y axis = π/4

Now, Using the formula,

Cos²α + Cos²β + Cosγ = 1

where α is the angle which vector makes with x-axis , β is the angle which vector makes with y-axis and γ is the angle which vector makes with z-axis.

∴ Cos²π/3 + Cos²π/4 + Cos²γ = 1

1/4 + 1/2 + Cos²γ = 1

Cos²γ + 3/4 = 1

Cos²γ = 1/4

Cosγ = 1/2

Cosγ = Cosπ/3

∴ γ = π/3

Hence, angle of the unit vector with z-axis is 60°.

Now, for components,

Component along x-axis = aCosα = 1 × 1/2 = 1/2

Component along y-axis = aCosβ = 1 × 1/√2 = 1/√2

 Component along z-axis = aCosγ = 1 × 1/2 = 1/2

Hope it helps.