Suppose you start at the origin, move along the x-axis a distance of 9 units in the positive direction, and then move downward a distance of 2 units. What are the coordinates of your position? (x, y, z) = ( )
Use the given points to answer the following questions. A(−3, 0, −6), B(4, 3, −7), C(2, 2, 2) Which of the points is closest to the yz-plane? A B C Which point lies in the xz-plane? A B C
Consider the point. (3, 4, 6) What is the projection of the point on the xy-plane? (x, y, z) = ( ) What is the projection of the point on the yz-plane? (x, y, z) = ( ) What is the projection of the point on the xz-plane? (x, y, z) = ( ) Draw a rectangular box with the origin and (3, 4, 6) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.
What does the equation x = 7 represent in R2? a point a line a plane a circle What does it represent in R3? a point a line a plane a circle Illustrate with sketches. Graph Graph Description Graph Graph Description Interactive 3D Graph Help Interactive 3D Graph Help Graph Graph Description Graph Graph Description Interactive 3D Graph Interactive 3D Graph
Find the lengths of the sides of the triangle PQR. (5, −2, −1), Q(9, 0, 3), R(11, −4, −1) |PQ| = |QR| = |RP| = Is it a right triangle? Yes No Is it an isosceles triangle? Yes No
Find an equation of the sphere with center (−2, 1, 7) and radius 8. What is the intersection of this sphere with the yz-plane?
Find an equation of the sphere that passes through the point (1, 6, 3) and has center (4, 3, −1).
Write the equation of the sphere in standard form. x2 + y2 + z2 +10x − 8y + 4z + 36 = 0 Find its center and radius. center (x, y, z) = ( ) radius
Describe in words the region of R3 represented by the equation. x2 + y2 = 49 Here x2 + y2 = 49 with no restrictions on, so a point in the region must lie on a circle of radius and center on the -axis, but it could be in any horizontal plane = k (parallel to the -plane). Thus, the region consists of all possible circles x2 + y2 = 49, = k and is therefore a with radius whose axis is the -axis.