Question

prove cos^2 alpha + cos^2 beta + cos^2 gamma = 1

please help...

Answer 1

We have to prove, cos²α + cos²β + cos²γ = 1

it is based on concept of 3D - geometry. you know, direction cosine of a line is defined as thr angle made by lines with the positive direction of coordinate axes.

let a line r is made α, β and γ with X - axis, y-axis and z - axis respectively.

now component of r along x-axis is |r|cosα

similarly, component of r along y-axis is |r|cosβ

and component of r along x-axis is |r|cosγ

so we can write it in vector form,

i.e., r = |r|cosα i + |r|cosβ j + |r|cosγ k

now |r| = √{|r|²cos²α + |r|²cos²β + |r|²cos²γ}

⇒|r| = |r|√{cos²α + cos²β + cos²γ}

⇒1 = cos²α + cos²β + cos²γ

hence proved

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Answer 2

$\huge\bold\purple{Answer:-}$

it is based on concept of 3D - geometry. you know, direction cosine of a line is defined as thr angle made by lines with the positive direction of coordinate axes.

let a line r is made α, β and γ with X - axis, y-axis and z - axis respectively.

now component of r along x-axis is |r|cosα

similarly, component of r along y-axis is |r|cosβ

and component of r along x-axis is |r|cosγ

so we can write it in vector form,

i.e., r = |r|cosα i + |r|cosβ j + |r|cosγ k

now |r| = √{|r|²cos²α + |r|²cos²β + |r|²cos²γ}

⇒|r| = |r|√{cos²α + cos²β + cos²γ}

⇒1 = cos²α + cos²β + cos²γ

hence proved