Please solve only number 4 showing work. 1 : 2x -y + 2z = 3, and 2 : 2x + 2y - z
ID: 3342858 • Letter: P
Question
Please solve only number 4 showing work.
1 : 2x -y + 2z = 3, and 2 : 2x + 2y - z = 3. Find the equation of the plane 3 in R3, which passes through P = (1, 1, 1) and is orthogonal to the planes 1 and 2. Find the volume of the parallelepiped U: t1a1 + t2a2 + t3a3 : 0 t1, t2, t3 1 with sides the vectors a1 = (5, 2, 4), a2 = (2, 1, 2), and a3 = (5, 1, 3). Parameterize the plane curve X : x4 + y4 = x2y + xy2. Find the velocity vector v(t) = X'(t) and the acceleration v X"(t) at each point X(t) of the curve X : X(t) = (2cos(t), sin(t), t), 0 t 2 pi,Explanation / Answer
use formula: A "dot" (B x C)
The volume of a parallelepiped is defined as the absolute value of the determinant of this matrix:
| a1 a2 a3 |
| b1 b2 b3 |
| c1 c2 c3 |
=
| 5 2 5|
| 2 1 1 |
| 4 2 3 |
You just have to memorize that volume of parallelepiped = abs. value of determinant of a 3x3 matrix where the rows are just the vectors you are given. Then you have to know how to take the determinant of a 3x3 matrix. You should memorize and use the Rule of Sarrus for this (you can find this on wikipedia). Taking the determinant of a matrix is used in lots of places (volume of a parallelepiped is just one example) so it is more important to learn how to do this. Basically, you draw out the 5 columns that are shown on wikipedia and add the products along the solid lines and subtract the products along the dashed lines.
So the answer is:
5(3-2)-2(6-4)+5(4-4)
=5(1)-2(2)+0
=1