Analytic Geometry Answers

(4.3) Let ~u =< 0, 1, 1 >, ~v =< 2, 2, 0 > and w~ =< −1, 1, 0 > be three vectors in standard form.

(a) Determine which two vectors form a right angle triangle?

(b) Find θ := ~ucw~ , the angel between the given two vectors.

(4.4) Let x < 0. Find the vector ~n =< x, y, z > that is orthogonal to all three vectors

~u =< 1, 1, −2 >, ~v =< −1, 2, 0 > and w~ =< −1, 0, 1 >.

(4.5) Find a unit vector that is orthogonal to both ~u =< 0, −1, −1 > and ~v =< 1, 0, −1 >.

(4.1) Consider the point A = (−1, 0, 1), B = (0, −2, 3), and C = (−4, 4, 1) to be vertices of a

triangle ∆. Evaluate all side lengths of ∆.

(4.2) Let ∆ be the triangle with vertices the points P = (3, 1, −1), Q = (2, 0, 3) and R = (1, 1, 1).

Determine whether ∆ is a right angle triangle. If it is not, explain with reason, why?

(3.1) Find an expression for 1

2

||~u + ~v||2 +

1

2

||~u − ~v||2

in terms of ||~u||2 + ||~v||2

.

(3.2) Find an expression for ||~u + ~v|| 2 − ||~u − ~v||2

in terms of ~u · ~v

(3.3) Use the result of (3.2) to deduce an expression for ||~u + ~v||2 whenever ~u and ~v are orthogonal

to each other.

Determine proj~a

~u the orthogonal projection of ~u and ~a and deduce ||proj~a

~u|| for

(2.1) ~u =< −1, 3 >, ~a =< −1, −3 >;

(2.2) ~u =< −2, 1, −3 >, ~a =< −2, 1, 2 >.

B ​​Quadratics in Motion: (​7, 6)

C​ Towers of Trigonometry: (​2, 14) D​​ The Food Equation: (​5, 8) 

Every square on the graph equals 60 m in the park. Jane and Bob are going to design a circular Ride that goes through the points B, C and D. 

1. Help them calculate where the centre of the ride would be.

Let u=<-2,1,-1, V =< - 3,2 - 1> and w=<1,3,5>. Compute :

a.) u x w,

b.) u x (w x v) and (u x w) x v.

Determine in each case whether the given planes are parallel or perpendicular (2) (2) (a) x + y + 3z +10 - 0 and x + 2y - Z-1. (b) 3x - 2y + 2 - 6 - 0 and 4x +2y - 42 -0. (c) 3x + y +2 -1 = 0 and -x +2y +2 +3 = 0, (+0 (d) x - 3y + 2 + 1 - 0 and 3x - 4y +2 -1 -0. (2)

Assume that a vector ~a of length ||~a|| = 3 units. In addition, ~a points in a direction that is 135◦ counterclockwise from the positive x-axis, and a vector ~b in the xy-plane has a length ||~b|| = 1/3 and points in the positive y-direction. Find ~a · ~b.

Determine whether u and v are orthogonal vectors,make an acute or obtuse angle:

1.u=<1,3,-2>; v=<-5,3,2>

2.u=<1,-2,4>; v=<5,3,7>

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?