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Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Round to one decimal place.)

$ 5x + 2y + 3z = 2 $ , $ y = 4x - 6z $

The two planes are orthogonal

Vectors

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Johns Hopkins University

Harvey Mudd College

Idaho State University

Boston College

in the question they're asking to determine where the prints are parallel, perpendicular or neither? If neither then find the angle between them. The equations of the two planes are number 15 X plus two, Y plus three. Is that equal to two? And # two was equal to four X -6. said. So the first equation is in regular form and so really changing the second equation, we get four x minus Why- Exit is equal to zero. And the first condition is to check these two planes are parallel or not. So for this we have to compare the coefficients of each of the coordinates of the two planes for the experience and access respectively. So this is equal to five by four is equal to two x -1 is equal to three x -6. And after simplifying these ratios, we get The ratio is five x 4 is equal to -2 is equal to minus half. And so these three ratios are not equal. Hence these planes, I am not Berlin taking the second criteria of the question whether there's two planes I'm perpendicular or not. So in order to check whether these two planes are perpendicular. Okay, we have to find the dog products of the normal vectors of these two planes. So the normal vector of the first plane is equal to indicated by the direction numbers, That is 5- three. And the Normal vector of the second plane and two is equal to the direction vectors are four -1 -6. Therefore, in order to find out dot product of these two normal vectors, It is always a skill er quantity, so this is equal to five into 4 -2, into one -6 into three, But this is equal to 20-2 -18, that is equal to 18 -18, that is equal to zero, and hence as the The product of the normal vectors of these two planes is equal to zero. Therefore, these planes are also gonna mhm and this is the required answer of the given question.