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My cat is standing on the plane $3x+2y+z = 4.$ Its head is at $(0,-1,-3)$ and the tip of its tail is at $(-4,-2,-2).$ Measured from the plane, what is the difference between the height of my cat's head and my cat's tail? Enter your answer in the form $\frac{a \sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are all positive integers.

 May 8, 2022
 #1
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The answer is 10*sqrt(5)/5.

 May 8, 2022
 #2
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The normal vector of the plane is \(3\vec{i} + 2\vec{j} + \vec{k}\).

 

Now, we calculate the vector from (0, -1, -3) to (-4, -2, -2).

\(\vec v = (-4, -2, -2) - (0, -1, -3) = (-4, -1, 1) = -4\vec i - \vec j + \vec k\).

 

We project that onto the normal vector to get the normal component of the vector, and find the length of the normal component to find the height. Can you proceed from here?

 May 8, 2022

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