1. a. The points (3,-1,2) and (-1,3,-4) are the endpoints of a diameter of a sphere i. Determine the center and radius of the sphere. ii. Find an equation for the sphere. b. Given the vectors a=2i-j+2 k₂ b=3i+2j-k, c =i + 2 k i. Calculate: 2a (b-3c) ii. Determine the vector projection of e onto b. iii. Find the cosine of the angle between a and b. iv. Find a unit vector that is perpendicular to the plane determined by a and e 2. Given the planes P: 2(x-1)-(y+1)-2(=-2)=0, P₂: 4x-2y + 5z =3\ and the point Q: (-2,7,4). a. Determine whether P₁ and P₂ are parallel, coincident, perpendicular, or none of the preceding. b. Find an equation for the plane through Q which is parallel to P₁ c. Determine scalar parametric equations for the line through Q which is parallel to the line of intersection of P₁ and P₂ 3. The position of an object at time t is given by: r(t)=e^i+e'j-t√√2k, 0≤1<0⁰ a. Determine the velocity and the speed of the object at time t b. Determine the acceleration of the object at time t c. Find the distance that the object travels during the time interval 0≤t<** 4. a. A curve C in the plane is defined by the parametric equations: x=²+1, y = +1₂y=-1 i. Find the length of C from t=0 to 1= 2 ii. Find the curvature of C at t = 1
1. a. The points (3,-1,2) and (-1,3,-4) are the endpoints of a diameter of a sphere i. Determine the center and radius of the sphere. ii. Find an equation for the sphere. b. Given the vectors a=2i-j+2 k₂ b=3i+2j-k, c =i + 2 k i. Calculate: 2a (b-3c) ii. Determine the vector projection of e onto b. iii. Find the cosine of the angle between a and b. iv. Find a unit vector that is perpendicular to the plane determined by a and e 2. Given the planes P: 2(x-1)-(y+1)-2(=-2)=0, P₂: 4x-2y + 5z =3\ and the point Q: (-2,7,4). a. Determine whether P₁ and P₂ are parallel, coincident, perpendicular, or none of the preceding. b. Find an equation for the plane through Q which is parallel to P₁ c. Determine scalar parametric equations for the line through Q which is parallel to the line of intersection of P₁ and P₂ 3. The position of an object at time t is given by: r(t)=e^i+e'j-t√√2k, 0≤1<0⁰ a. Determine the velocity and the speed of the object at time t b. Determine the acceleration of the object at time t c. Find the distance that the object travels during the time interval 0≤t<** 4. a. A curve C in the plane is defined by the parametric equations: x=²+1, y = +1₂y=-1 i. Find the length of C from t=0 to 1= 2 ii. Find the curvature of C at t = 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1.
a. The points (3,-1,2) and (-1,3,-4) are the endpoints of a diameter of a sphere
i. Determine the center and radius of the sphere.
ii. Find an equation for the sphere.
b. Given the vectors a=2i-j+2 k₂ b=3i+2j-k, c =i + 2 k
i. Calculate: 2a (b-3c)
ii. Determine the vector projection of e onto b.
iii. Find the cosine of the angle between a and b.
iv. Find a unit vector that is perpendicular to the plane determined by a and e
2. Given the planes P: 2(x-1)-(y+1)-2(=-2)=0, P₂: 4x-2y + 5z =3\ and the point Q: (-2,7,4).
a. Determine whether P₁ and P₂ are parallel, coincident, perpendicular, or none of the preceding.
b. Find an equation for the plane through Q which is parallel to P₁
c. Determine scalar parametric equations for the line through Q which is parallel to the line of
intersection of P₁ and P₂
3. The position of an object at time t is given by: r(t)=e^i+e'j-t√√2k, 0≤1<0⁰
a. Determine the velocity and the speed of the object at time t
b. Determine the acceleration of the object at time t
c. Find the distance that the object travels during the time interval 0≤t<**
4.
a.
A curve C in the plane is defined by the parametric equations: x=²+1, y =
+1₂y=-1
i. Find the length of C from t=0 to 1= 2
ii. Find the curvature of C at t = 1
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