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Find the acute angle between the plane 5x − 4y + 7z − 13 = 0 and the y-axis. - Mathematics

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Find the acute angle between the plane 5x − 4y + 7z − 13 = 0 and the y-axis.

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Solution

The equation of the y-axis is

`(x−0)/0=(y−0)/1=(z−0)/0`

The direction ratios of the y-axis are 0, 1, 0.

The equation of the given plane is 5x − 4y + 7z − 13 = 0.

So, the direction ratios of the normal to the plane are 5, −4, 7.

Let θ be the acute angle between the given plane and the y-axis.

`therefore sin theta=|(a_1a_2+b_1b_2+c_1c_2)/(sqrt(a_1^2+b_1^2+c_1^2)sqrt(a_2^2+b_2^2+c_2^2))|`

`=> sin theta=|(0xx5−4xx1+0xx7)/(sqrt(0+1+0)sqrt(25+16+49))|`

`=|-4/(3sqrt10)|`

`=>theta=sin^-1(4/3sqrt10)`

Hence, the acute angle between the given plane and the y-axis is `sin^-1(4/3sqrt10)`

Concept: Angle Between Line and a Plane
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