Analytic Geometry Answers

Compute and find a relation between the expressions

(8.1) ~u · (~v × w~ ), w~ · (~u × ~v), and ~v · (w~ × ~u)

Find an expression for :

(8.2) ~u · (~v × w~ ) and ~u × (~v × w~ )

(8.3) Find the side lengths and angles of the triangle with vertices the tips the vectors in Question 6 above.

(8.4) Use ~u and ~v from Question 6 to find the area of the parallelogram formed by ~u and ~v.

(8.5) Use ~u, ~v and w~ from Question 6 to find the volume of the parallelepiped with edges deter- mined by the three vectors ~u, ~v and w~ 

We assume given a plane U passing by the tip of the vectors ~u =< −1, 1, 2 >, ~v =< 2, −1, 0 > and w~ =< 1, 1, 3 >.

(6.1) Find the dot products ~u · ~v and w~ · ~v

(6.2) Determine whether or not there is a vector ~n that is perpendicular to U. If yes, then find the vector ~n. Otherwise explain why such a vector does not exist?

Consider the vectors ~u =< −2, 2, −3 >, ~v =< −1, 3, −4 >, w~ =< 2, −6, 2 > and the points A(2, 6, −1) and B(−3, −5, 7). Evaluate

(5.1) The distance between the two points.

(5.2) ||2~u − 3~v + (1) 1 2w~ ||.

(5.3) The unit vector in the direction of w~ .

(5.4) Suppose ~u; ~v and w~ are vectors in 3D, where ~u = (u1, u2, u3) ; ~v = (v1, v2, v3) and w~ =(w1, w2, w3).

Express (~u × ~v) · w~ as a determinant. 

Let P = (1, 2, 3) and Q = (−5, −1, 12).

(4.1) Find the midpoint of the line segment connecting P and Q.

(4.2) Find the point on the line segment connecting P and Q that is 2/3 of the way from P to Q.

(4.3) Let P = (−1, 5, 2). If the point (0, −2, 3) is the midpoint of the line segment connecting P and Q, what is the point Q?

(3.1) Find the area of the triangle with the given vertices A(1, 3), B(−3, 5), and C with C = 2A.

(3.2) Use (3.1) to find the coordinates of the point D such that the quadrilateral ABCD is a parallelogram. 

Use the fact that :

0= x y 1

a1 b1 1

a2 b2 1

to determine the equation of the line passing through the distinct points (a1, b1) and (a2, b2), where |·| stands for det(·), the determinant. 

Identify and trace the conicoid y² +z² = x. Describe its sections by the planes x = 0, y = 0 and z = 0

Find the transformation of the equation 12x² −2y² +z² = 2xy if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are 1,−3,0; 3,1,0; 0,0,1.

Find the equations of the spheres which pass through the circle x² +y² +z² = 9,2x+2y−7 = 0 and the touch the plane x−y +z +3 = 0

Obtain the equation of the conic, a focus of which lies at (2,1), the directrix of which is x+y = 0 and which passes through (1,4). Also identify the conic.