# Are the planes #x+y+z=1# , #x-y+z=1# parallel, perpendicular, or neither? If neither, what is the angle between them?

We can find the angle between two planes, by finding the angle between the normals to the plane:

When given the equation of a plane in Cartesian form:

The normal vector to the plane is:

Let:

Then:

We can find the angle between these normals using the Dot Product:

If the planes were perpendicular, the the angle between them would be

If the planes were parallel, the the angle between them would be

Plot:

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To determine whether the planes (x+y+z=1) and (x-y+z=1) are parallel, perpendicular, or neither, you can compare their normal vectors.

The normal vector of a plane with equation (Ax + By + Cz = D) is ((A, B, C)).

For the plane (x+y+z=1), the normal vector is ((1, 1, 1)).

For the plane (x-y+z=1), the normal vector is ((1, -1, 1)).

To determine their relationship, calculate the dot product of the normal vectors.

If the dot product is zero, the planes are perpendicular.

If the dot product is non-zero and the normal vectors are not parallel, the planes are neither parallel nor perpendicular.

If the dot product is non-zero and the normal vectors are parallel, the planes are parallel.

In this case, the dot product of ((1, 1, 1)) and ((1, -1, 1)) is (1*1 + 1*(-1) + 1*1 = 1 - 1 + 1 = 1), which is not zero.

Also, the normal vectors are not parallel because they are not scalar multiples of each other.

Therefore, the planes (x+y+z=1) and (x-y+z=1) are neither parallel nor perpendicular.

To find the angle between them, you can use the formula for the angle between two vectors:

[ \cos(\theta) = \frac{{\mathbf{a} \cdot \mathbf{b}}}{{|\mathbf{a}| |\mathbf{b}|}} ]

where (\mathbf{a}) and (\mathbf{b}) are the normal vectors of the planes.

Then, find the angle (\theta) using the inverse cosine function.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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