Analytic Geometry Answers

Find the Cartesian equations of the lines or curves whose polar equations are (a) r = a (b) r = a cos 2θ

(c) r2 = a2 sin 2θ (e) r = a sec(θ − α)

show that for any three vectors a,b,c

a×(b×c)+b×(c×a)+c×(a×b)=0

suppose A=3i-j-2k and B=2i+3j+k,Find |AxB|,(A+2B)x(2A-B),(A+B)x(A-B)

A software designer is mapping the streets for a new racing game. All of the streets are depicted as either perpendicular or parallel lines. The equation of the lane passing through A and B is -7x + 3y = -21.5. What is the equation of the central street PQ?

Consider the vectors

u = (1, 0) and v = (0, 1).

7.1 Determine cos θ, where θ is the angle between u and v. (4)

7.2 Determine the area of the parallelogram determined by u and v.

Let L1 and L2 be lines defined by

x = w0 + su, s ∈ R

and y = w1 + tv, t ∈ R, respectively.

Show that L1and L2 are parallel if and only if u = kv for some k ∈ R. (8)

8.2 Find the plane that passes through the point (2, 4, −3) and is parallel to

the plane −2x + 4y − 5z + 6 = 0. (4)

8.3 Find the line that passes through the point (2, 5, 3) and is perpendicular to the plane

2x − 3y + 4z + 7 = 0. (4)

8.4 Find an equation of the plane passing through the point(−2, 3, 4) and is perpendicular to the line

passing through the points (4, −2, 5) and (0, 2, 4).

If \\(a=3i-2j+k\\), \\(b=2i-4j-3k\\) and \\(c=-i+2j+2k\\), find the magnitude of \\(a+b+c\\)

Calculate the area of a triangle whose vertices are given by (3, -1, 2), (1, -1, 2)

and (4, -

2, 1).

If \\(a=3i-2j+k\\), \\(b=2i-4j-3k\\) and \\(c=-i+2j+2k\\), find the magnitude of \\(a+b+c\\)

Answer the questions and show complete solutions.


6. A plane through the origin is perpendicular to the plane 2