Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2. r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2π Then the vector-valued function describes a tilted circle (a circle in a plane that is not parallel to one of the coordinate planes). The center of the circle is O(0, 0, 0). (a) Show that |r(t)| is constant and determine the constant. This is the radius of the circle. (b) Find an equation for the plane that contains the circle by doing the following: (i) Find three points on the circle. (ii) Use the three points to find a normal vector to the plane containing the three points. (iii) Find an equation for the plane. Check: does the plane contain the center of the circle? (iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2. r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2π Then the vector-valued function describes a tilted circle (a circle in a plane that is not parallel to one of the coordinate planes). The center of the circle is O(0, 0, 0). (a) Show that |r(t)| is constant and determine the constant. This is the radius of the circle. (b) Find an equation for the plane that contains the circle by doing the following: (i) Find three points on the circle. (ii) Use the three points to find a normal vector to the plane containing the three points. (iii) Find an equation for the plane. Check: does the plane contain the center of the circle? (iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2. r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2π Then the vector-valued function describes a tilted circle (a circle in a plane that is not parallel to one of the coordinate planes). The center of the circle is O(0, 0, 0). (a) Show that |r(t)| is constant and determine the constant. This is the radius of the circle. (b) Find an equation for the plane that contains the circle by doing the following: (i) Find three points on the circle. (ii) Use the three points to find a normal vector to the plane containing the three points. (iii) Find an equation for the plane. Check: does the plane contain the center of the circle? (iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2. r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2π Then the vector-valued function describes a tilted circle (a circle in a plane that is not parallel to one of the coordinate planes). The center of the circle is O(0, 0, 0). (a) Show that |r(t)| is constant and determine the constant. This is the radius of the circle. (b) Find an equation for the plane that contains the circle by doing the following: (i) Find three points on the circle. (ii) Use the three points to find a normal vector to the plane containing the three points. (iii) Find an equation for the plane. Check: does the plane contain the center of the circle? (iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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