I have an assignment for Vectors and it is basically as follows just as reference.
Please create a vector and Cartesian equation for a plane (pi1) that is >based >on your birthday. Your birthday plane must include the points: (mm, dd, >yy), (dd, yy, mm), (yy, mm, dd).
Create an vector and Cartesian equation for a 2nd plane (Pi2) that >celebrates starting Calculus at Sir William Mulock S.S. This should include the >Liebniz birthday vector [1, 7, 46](we use square brackets to represent a >vector), the point of Newton's birthday (4, 1, 43) and the point at which our >school building opened to students (3, 12, 1)
Find a plane (Pi3) that intersects with your plane and the Mulock Calculus >plane at exactly 1 point. The normal of (Pi3) may not contain any zeros. The >point of intersection needs to contain the coordinate x=99.
- Give the vector and Cartesian equation of your newly created plane. (Pi3)\
.4. Show that there is one point of intersection between the three planes and > what that point is. . 5. Find the angle between the plane you created (Pi3) and your birthday plane >>>(Pi1)
- Find the shortest distance from the point the school opened on (03, 12, 01) >to your new plane (Pi3).
I'm currently having trouble figuring out the way to get Pi3, like how would we find the plane knowing the point all 3 planes intersect is at (99, b, c) and normal cannot have a 0
I would appreciate any help for 3 to 6, I already did 1-2 which were easy. It is very time consuming assignment.
Thanks in advance
It doesn’t seem like this should be terribly time-consuming, although some of the computations are a bit tedious.
Going back to part 1, you can save yourself some work by observing that the three points are obtained by successively multiplying by the matrix $$\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}$$ (or its transpose if you’re working with row vectors), which represents a 120° rotation about the axis $(1,1,1)^T$. All three points lie in a plane that’s orthogonal to this axis, so the equation of the plane can be obtained quickly and easily by using the point-normal form: $$x+y+z-mm-dd-yy=0.$$ You might equivalently note that the coordinates of the three points are cyclic permutations of each other, so the equation of the plane must be invariant with respect to cyclic permutations of the variables $x$, $y$ and $z$, therefore their coefficients must be equal.
Part 2 does require a bit of computation, but while tedious, it’s not terribly difficult. For part 3, the two planes $\mathbf\pi_1$ and $\mathbf\pi_2$ intersect in a line that’s orthogonal to the normals of the two planes. For $\mathbf\pi_3$ to intersect this line in only a single point, it cannot contain the line. An easy way to ensure this is to choose the line’s direction vector $\mathbf v$ as the normal of this plane. (This will turn out not to have any coordinates equal to zero, so it satisfies the other condition of the problem.) If you write the line’s equation in the parametric form $\mathbf p_0+\lambda\mathbf v$, you can then solve for the value of $\lambda$ that results in an $x$-coordinate equal to $99$ and plug that point into the point-normal form of the equation for a plane. Alternatively, plug the point $(99,a,b)$ into the equations of $\mathbf\pi_1$ and $\mathbf\pi_2$ and solve the resulting system of equations for $a$ and $b$ to get the point to use for the point-normal equation. Again, somewhat tedious because of the largish numbers that might appear, but it shouldn’t take very long to perform this computation, either. For the vector form of the equation of $\mathbf\pi_3$, you’ve already got two linearly independent vectors that parallel this plane, so once you’ve computed the above point, you can write that equation down without further ado.
The above construction guarantees a single point of intersection among the three planes, so I should think that satisfies part 4. You also know that $\mathbf\pi_1$ and $\mathbf\pi_3$ are orthogonal by construction, so that takes care of part 5 with no additional work. Finally, for part 6 simply use the formula for the distance between a point and plane.