Question

Show that a vector


`u = x _(1)i + y_(1)j + z_(1)k`
with direction cosines (cos α, cos β, cos gamma) can be written as

`u = |u| ( \cos \alpha i + \cos\beta j + \cos \gamma k)`
​

Answer 1

u = |u|(cos∝+cosβ+cosγ)

Explanation:

Explanation

Proof:-

Given a vector u = x₁ i + y₁j +z₁k

let O X, OY, O Z be the positive co-ordinate axes

P(x₁,y₁,z₁) be any point in the space

Let OP makes angles α,β,γ with co-ordinate axes OX , OY ,OZ .

The angle α,β,γ are known as direction angles and cosine of the angle

l =cosα , m= cosβ , n=cosγ

The perpendicular PA,PB,PC are drawn co-ordinate axes OX,OY,OZ respecctively

InΔOAP , ∠A =90° , cos∝ =
`(x)/(r)`

x₁ = rcos∝

InΔOBP , ∠B =90° , cosβ =
`(y)/(r)`

y₁ = rcosβ

InΔOCP , ∠C =90° , cosγ =
`(z)/(r)`

z₁ = rcosγ

Given u = x₁ i + y₁j +z₁k

|u| =
`\sqrt{(x_(1))^(2) +(x_(2) )^(2) +(x_(3) )^(2) }`

Therefore u = x₁ i + y₁j +z₁k

u = |u|(cos∝+cosβ+cosγ)

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