Show that a vector
with direction cosines (cos α, cos β, cos gamma) can be written as

 u = |u|(cos∝+cosβ+cosγ)

Step-by-step 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∝ =

                                 x₁ = rcos∝

InΔOBP , ∠B =90° , cosβ =

                                 y₁ = rcosβ

InΔOCP , ∠C =90° , cosγ =

                                z₁ = rcosγ

Given   u = x₁ i + y₁j +z₁k

      |u| =

Therefore  u = x₁ i + y₁j +z₁k

               u = |u|(cos∝+cosβ+cosγ)

answer: a.   one is a factor of every whole number since every number is divisible by itself.