# How do you find the angle between the planes #x + 2y - z + 1 = 0# and #x - y + 3z + 4 = 0#?

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Angle between two planes is given by the angle of their normal vectors (caracteristic). The caracteristic vectors are given by coefficients of x, y and z in equation plane

By other hand we know that dot product vector is given by

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To find the angle between two planes given by their equations in the form Ax + By + Cz + D = 0, you can use the formula:

cos(θ) = |A₁A₂ + B₁B₂ + C₁C₂| / √(A₁² + B₁² + C₁²) * √(A₂² + B₂² + C₂²)

Where (A₁, B₁, C₁) and (A₂, B₂, C₂) are the normal vectors to the planes.

For the planes x + 2y - z + 1 = 0 and x - y + 3z + 4 = 0, the normal vectors are (1, 2, -1) and (1, -1, 3) respectively.

Plugging these values into the formula:

cos(θ) = |1*1 + 2*(-1) + (-1)*3| / √(1² + 2² + (-1)²) * √(1² + (-1)² + 3²)
= |1 - 2 - 3| / √(1 + 4 + 1) * √(1 + 1 + 9)
= |-4| / √6 * √11
= 4 / √(6 * 11)
= 4 / √66

Therefore, the angle between the planes is arccos(4 / √66), which is approximately 78.24 degrees.

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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.

- A triangle has corners at #(4 ,7 )#, #(2 ,6 )#, and #(3 ,8 )#. How far is the triangle's centroid from the origin?
- Circle A has a center at #(5 ,-4 )# and a radius of #7 #. Circle B has a center at #(-6 ,-8 )# and a radius of #5 #. Do the circles overlap? If not, what is the smallest distance between them?
- A triangle has corners at #(4 ,3 )#, #(2 ,6 )#, and #(7 ,1 )#. How far is the triangle's centroid from the origin?
- What is the perimeter of a triangle with corners at #(6 ,0 )#, #(5 ,2 )#, and #(5 ,4 )#?
- A line passes through #(2 ,5 )# and #(0 ,1 )#. A second line passes through #(7 ,1 )#. What is one other point that the second line may pass through if it is parallel to the first line?

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