7. Problems

[not to hand in unless assigned]

Problem B-1. Suppose $P,Q,R,S$ are vertices of a parallelogram in R$^2$, with the line $PQ$ parallel to $RS$ and $SP$ parallel to $QR$. Find an algebraic expression for $S$ in terms of $P$, $Q$, and $R$. (Notice that if $P$, $Q$, $R$ are given, there's only one possible place for $S$.)

Problem B-2. Find the equation of the plane through points $P = (1,2,1)$, $Q = (3,4,3)$, $R = (2,2,3)$ in R$^3$.

Method: Find the plane through $P$ with normal $(Q-P) \times (R-P)$.

Problem B-3. Find the angle between the planes $x+2y+z = 0$ and $3x+y+z=0$.

Method: Find the angle between their normals. The angle between two planes can be looked at so that $0 \leq \theta \leq 90^\circ$, so adjust your answer if necessary to achieve this.

Problem B-4. Find the line of intersection of the two planes in the preceding problem.

Method: The line wanted is perpendicular to both normals and goes through the origin.

Problem B-5. Find the line of intersection of the two planes $x + 2y + z - 3 = 0$ and $3x + y + z - 4 = 0$.

Method: The line wanted has the same direction as in the preceding problem, but does not go through the origin. To find a point on the line, one way is to pick an arbitrary value for one of $x$, $y$, and $z$ and solve the resulting two-variable equations simultaneously to get values for the other two variables.

Problem B-6. In R$^2$, find the cosine of the angle between the lines $3x + 4y + 7 = 0$ and $4x + 3y - 2 = 0$. (Do like the similar problem above for planes in R$^3$.)

Problem B-7. If $P,Q,R$ are as in Problem B-, are the points $A,B$ on the same side of the plane through $P,Q,R$, where $A = (1,2,3)$ and $B = (1,2,5)$?

Method #1: Find the equation of the plane in the form $ax + b y + cz + d = 0$, and see whether the left-hand side has the same sign when evaluated at $A$ and at $B$.

Method #2: Let N$= (Q-P) \times (R-P)$ and see whether the angles between $A-P$ and N and between $B-P$ and N are both between $0 ^\circ$ and $90 ^\circ$ or both between $90 ^\circ$ and $180 ^\circ$.

Problem B-8. Find the vector projection of v$= (1,1,1)$ on the line in the direction of w$= (1,2,2)$.

Problem B-9. Find the vector that is the projection of the vector v$= (1,1,1)$ on the plane $x + 2y + 2z = 0$.

Problem B-10. Find the signed angle between $(1,2)$ and $(2,-1)$ using the method of §. For atan2, just evaluate it as a computer would.