Analytic Geometry Answers

Suppose u; v and w are vectors in 3 space, where u = (u1;u2; u3) ; v = (v1; v2; v3) and w = (w1;w2;w3) :
Express (u x v) x w as a determinant

If v∈R2 lies in the first quadrant and makes an angle π/4 with the positive x-axis and ||v||=2, then, Select one:


a. v=⟨2√2, 2√2⟩


b. v=⟨2, 2√3⟩


c. v=⟨√2, √2⟩


d. v=⟨2, −2√3⟩


e. v=⟨2√2, −√2⟩

Let u = (4,2,-1), v = (3, 1, 1) and w = (0, 2, 1). Compute the following:
(i) 2v - 3w -u
(ii) u(w+v)
(iii) ||u . w ||
(iv) the orthogonal projection of u on w
(v) the vector component of u orthogonal to w

Which of the following is an ellipse? More than one answer may be correct.
A. x^2÷3^2+y^2÷3^2=1
B. x^2=−y
C. y^2=4(1−x^2)
D. None of the above

Find the equation of the normal to the solid 2x^2 −y^2 +8z^2 = 11 at a point where it intersects the line x−3 = z =(y+1)/2.

A plane passes through (a,b, c) and cuts the axes in A,B,C, respectively, where
none of these points lie on the origin O. Show that the centre of the sphere OABC
satisfies the equation a/x +b/y +c/z = 2.

The normals at any point P of the ellipsoid x^2/9 +y^2+4 +z^2 = 1 meet the coordinate planes in Q1,Q2,Q3, respectively. Show that PQ1 : PQ2 : PQ3 :: 9 : 4 : 1.

Check whether the points (1,−1,−2),(1,−4,2),(3,0,2),(4,−3−2) are coplanar
or not. If they are coplanar, write the equation of the plane they pass through.
Otherwise, change the coordinates of one of the points so that they become
coplanar. In this case, find the plane passing through them.

Derive the equation (23) at page 42 of Unit 2, which represents the polar equation of a conic when the directrix L corresponding to a focus F is taken to the right of F.

Find the equation of the hyperbola with vertices (1,−4) and (1,4), and foci at(1,−6) and (1,6).