Question

The line makes angles α, β and γ with x-axia and z-axis respectively then cos 2α + cos 2β + cos 2γ is equal to

(a) 2
(b) 1
(c) -2
(d) -1​

Answer 1

Explanation:

`\large\underline{\sf{Solution-}}`

Given that lines makes an angle α, β, γ with x - axis, y - axis and z - axis respectively.

So, By definition of direction cosines,

`\rm :\longmapsto\:l = cos \alpha`

`\rm :\longmapsto\:m = cos \beta`

`\rm :\longmapsto\:n = cos \gamma`

So,

`\rm :\longmapsto\: {l}^(2) + {m}^(2) + {n}^(2) = 1`

`\rm :\longmapsto\: {cos}^(2) \alpha + {cos}^(2) \beta + {cos}^(2) \gamma = 1`

On multiply by 2 on both sides we get

`\rm :\longmapsto\: 2{cos}^(2) \alpha + 2{cos}^(2) \beta + 2 {cos}^(2) \gamma = 2`

can be further rewritten as

`\rm :\longmapsto\: 2{cos}^(2) \alpha - 1 + 1 + 2{cos}^(2) \beta - 1 + 1 + 2 {cos}^(2) \gamma - 1 + 1 = 2`

`\rm :\longmapsto\: (2{cos}^(2) \alpha - 1)+ (2{cos}^(2) \beta - 1)+ (2 {cos}^(2) \gamma - 1) + 3= 2`

`\rm :\longmapsto\:cos2 \alpha + cos2 \beta + cos2 \gamma + 3= 2`

`\red{ \bigg\{ \sf \: \because \: cos2x = {2cos}^(2)x - 1 \bigg\}}`

`\rm :\longmapsto\:cos2 \alpha + cos2 \beta + cos2 \gamma= 2 - 3`

`\rm :\longmapsto\:cos2 \alpha + cos2 \beta + cos2 \gamma= - 1`

Hence,

`\bf\implies \:\boxed{\tt{ \: cos2 \alpha + cos2 \beta + cos2 \gamma = - 1 \: }}`

So, option (d) is correct.

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MORE TO KNOW

Direction cosines of a line segment is defined as the cosines of the angle which a line makes with the positive direction of respective axis.

The scalar components of unit vector always give direction cosines.

The scalar components of a vector gives direction ratios.