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SB
Numerade Educator

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Problem 53 Medium Difficulty

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

$ x + 2y - z = 2 $ , $ 2x - 2y + z = 1 $

Answer

Angle between the two planes is $114.1^{\circ}$

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Video Transcript

in the question they're asking to determine whether the planes are parallel, perpendicular or neither. If neither then find the angle between them. The two equations of the planes are number one express two by minus that equal to two and number 22 x minus two. Y pass that equal to one. The first condition of taking whether these planes are parallel or not. We have to compare the coefficients, The coordinates of these two planes for the X, Y and Z coordinates respectively. That is one way to equal to two by minus two equal to minus one by one. That is, it will do half. That is not equal to -1 and not equal to minus one as all three three shoes are not equal. Mhm. So Mhm. These planes are not Caroline. 2nd part is checking. These planes are perpendicular are not. What is it? So for this we have to find the normal vectors of these two planes. So normal victor, one is equal to 1, -1. and the normal Vector two for the second plane is equal to 2 -2. 1. Therefore we have to find the dot product of these two. Normal vectors, if it is equal to zero then these planes are perpendicular. Otherwise not therefore, and one vector. And to vector is equal to two minus 4 -1, that is equal to -3 and this is not equal to zero. Therefore these planes are not what particular to each other. And the third point is if they are neither of this, if neither, then we have to find them anger between from these planes. Yeah. So in order to find the angle between these planes the formula is cost Theta is equal to the skill er one today of and two victor dot in one victor who divided by skill er off in to victor Multiplied by a scalar of in one victor. Therefore cost 3 to is equal to putting the values of the normal victims. And these killers we get The numerator to be to -4 -1 divided by route under one squared plus two squared plus one square into he wrote under two square place to square place one square it won't. So casita is equal to -3 divided by Route Under six. In 2 route under nine this is equal to minus three divided by route under six. Multiplied by three and three and 3 from new murder and no Minuto gets canceled. So cost three to Is equal to -1 by Route six. So therefore tita Is equal to cause inverse of -1. by Route six. This value of Theta is equal to 114.07°. So in the question they have access to round The decimal digit up to one digit. So this is equal to 114.1 degree. Yeah And this is the angle between the two planes which is really quite answer of the given question. Just one