1. Lines in R1. (a) Draw the following three parallel lines in the Cart.hm plane: + 23 = —5, g + 29 = 0, z + 2y = 5. (b) Fill in the blanks: The equation as+ by = c represents all. in 112 that is perpendicidar to the vector and contains the point . [There are infinitely many corn. answem.] (c) Fill in the blanks: The two line am + by = e and + ry = d are perpendicular if and ordy if . They are parallel if and only J .
2. Two Equations in Two Unknowns. The following vector equation with two unknowns is .uivaknt to a system of two linear equntions in two unknowns:
– (-20 (0 = (30 –
y = 3,
2x + SY = 4.
(a) Solve the system to had m and y. (b) Let u= (-1,2) and v= (1,0). Draw your solution to the vector equation, using copies of ti and v to get from the might to the point (3.4). (Strang calls this the rolumn (c) Draw your solution to the linear system . the intersection of two lines. (Strang calk this the row pzturr.)
a. Planes In IP. (a) Fill in the blanks: The equation as + by + m = d represents a plane in le that k perpendicular to the vector and contains the point . (There are infinitely many corr. answers] (b) Fill in the blanks: The two lines as + by + cv = d and dz +crz = ‘11 are perpendieulm if and only if . They are parallel if and only if . (c) Fill in the blanks: The intersection of two planes in RI is probably a . The intersect.1 of three Man. N le in probably a
4. Throe Equations in Three Unknowns. Consider the following system of two linear equatiorw in three unknowns: fx+ y+2z=0, x + 2y — z = 0. (a) This system represems the intersection of two planes. Express the solution as parametrized line itv : for some direction vector v. [Hint: Let s = t be a free parameter] (b) Use your answer from part (a) to find some vector (x,y, z) that is simultaneously per-pendicular to both (1,2, —1) sod (1, 1,2). (There are infinitely many correct ans..] (c) Compute the inteiseeticm of the km from part (a) with the thind piano X* vice= —1.
(d) Finally, compute the solution 1 of the following vector equation: ( 2)+ s ( —1 ) = ( 0.
5. Hyperplanes in IP. (a) Fill in the blanks: The equation omi + ooze + • • • + = b rePrents a 3. _- dimensional shape in -dimensional space. This shape is called a hverplane. (b) Fill in the bleak: If m < ra then the intersection of m hyperplanes hl mdimensional space probably has dimension (0) Fill in the bleak: If m 5 It then the solution to a system of m linear equations in n unkuovms probably hen free parameters. (d) Fill in the blank: I( m > n then a system dm linear equations iu n unknew. probably