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Find the component form of aa¯ if it lies in YZ-plane makes 60° with positive Y-axis and a|a¯|=4. - Mathematics and Statistics

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Sum

Find the component form of `bar"a"` if it lies in YZ-plane makes 60° with positive Y-axis and `|bar"a"| = 4`.

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Solution

Let α, β, γ be the direction angles of `bar"a"`

Since `bar"a"` lies in YZ-plane, , it is perpendicular to X-axis

∴ α = 90°

It is given that β = 60°

∵ cos2α + cos2β + cos2γ = 1

∴ cos290° + cos260° + cos2γ = 1

∴ 0 + `(1/2)^2` + cos2γ = 1

∴ cos2γ  = `1 - 1/4 = 3/4`

∴ cos2γ = `+- sqrt3/2`

Unit vector along `bar"a"` is given by

`hat"a" = ("cos" alpha)hat"i" + ("cos"beta)hat"j" + ("cos"gamma)hat"k"`

`= 0.hat"i" + 1/2hat"j" + sqrt3/2hat"k"`

`= 1/2hat"j" +- sqrt3/2hat"k"`

∴ `bar"a" = |bar"a"|hat"a" = 4(1/2hat"j" +- sqrt3/2hat"k")`     .....[∵ `|bar"a"| = 4`]

∴ `bar"a" = 2hat"j" +- 2sqrt3hat"k"`

Concept: Vectors and Their Types
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