c Dr Oksana Shatalov, Spring 2014
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Spring 2014 Math 251
Week in Review 1
courtesy: Oksana Shatalov
(covering Sections 11.1-11.4 )
11.1: Three-dimensional Coordinate System
Key Points
• The distance between the points P (x1 , y1 , z1 ) and Q(x2 , y2 , z2 ) is
p
|P Q| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
• Equation of a sphere (x − a)2 + (y − b)2 + (z − c)2 = r2 (completing the square)
Examples
1. Describe in words the following regions in R3 represented by the equation:
(a) y = −1
(b) y 2 + z 2 = 1
(c) xy = 0
2. Find the center and radius of the sphere x2 + y 2 + z 2 − 6x + 10y − 2 = 0.
3. Describe the following sets in R3 .
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(a) {(x, y, z)| |z| ≤ 7}
(b)
(x, y, z)| x2 + z 2 ≤ 1, − 2 ≤ y ≤ 5
(c)
(x, y, z)| x2 + y 2 + z 2 ≤ 10, z ≥ 0
4. The sphere x2 + y 2 + z 2 = 25 intersects the plane y = 3 in a circle. What are the center and the
radius of the circle?
11.2: Vectors and the Dot Product in Three Dimensions
Key Points
• The vector a from the point P (x1 , y1 , z1 ) to Q(x2 , y2 , z2 ) is
a = hx2 − x1 , y2 − y1 , z2 − z1 i = (x2 − x1 )i + (y2 − y1 )j + (z2 − z1 )k.
p
• The magnitude or length of a = ha1 , a2 , a3 i is |a| = a21 + a22 + a23 .
• Unit vector: â =
a
|a|
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• Dot Product:
a · b = |a| · |b| cos θ = a1 b1 + a2 b2 + a3 b3 ,
where θ is the angle between a and b, 0 ≤ θ ≤ π.
a·b
|a|
a·b a
a·b
• Vector projection of vector b onto vector a: proja b =
=
a
|a|
|a|
|a|2
• Scalar projection of vector b onto vector a: compa b =
−→
• The work done by a force F in moving and object from point A to point B is given by W = F · D where D = AB is
the displacement of the object.
Examples
5. Find an equation of the sphere having the points A(1, 2, 3) and B(3, 4, 1) as endpoints of a diameter.
6. Given a = h1, 1, 1i and b = h−1, 2, −4i. Find a unit vector that has the same direction as the
vector 2a − b.
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7. What restrictions must be made on b so that the vector 2i + bj is orthogonal to vector 3i − 2j − k?
to vector k?
8. A force of F = −3i − j + 5k Newtons is applied to a particle that moves a distance of 6 meters in
the direction of the vector v = i − 2j + 2k. How much work is done?
11.3: Cross product
Key Points
• The cross product requires both of the vectors to be three dimensional vectors.
• The result of a dot product is a number and the result of a cross product is a VECTOR!!!
• To remember the cross product component formula use the fact that the cross product can be represented as the
determinant of order 3:
i
j
k a × b = a1 a2 a3 b1 b2 b3 • Properties:
a×a=0
a × b = −b × a
(αa) × b = a × (αb) = α(a × b),
a × (b + c) = a × b + a × c
(a + b) × c = a × c + b × c
α∈R
• |a × b| = |a| · |b| sin θ =the area of the parallelogram determined by a and b;
• a × b is orthogonal to both a and b;
• the direction of a × b is determined by “right hand” rule.
• akb
⇔ a × b = 0.
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• SCALAR TRIPLE PRODUCT a · (b × c) is a NUMBER.
• a · (b × c) = (a × b) · c
a1
• If a = ha1 , a2 , a3 i, b = hb1 , b2 , b3 i and c = hc1 , c2 , c3 i then a · (b × c) = b1
c1
a2
b2
c2
a3
b3
c3
• |a · (b × c)| = the volume of the parallelepiped determined by a, b, c.
Examples
9. State whether each expression is meaningful. If so, state whether it is a vector or a scalar.
(a) (a × (a + b)) · (b × (5a))
(b) (a · (a + b)) × (b × (5a))
10. Given a = h1, 0, −1i and b = h3, 1, 2i.
(a) Find all vectors orthogonal to both a and b.
(b) Find all unit vectors orthogonal to both a and b.
(c) What is the area of the parallelogram defined by a and b?
11. Do the points A(−2, 5, 5), B(1, 2, −1), and C(0, 3, 1) lie in a straight line?
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11.4: Equations of lines and planes
Key Points
• Parametric Equaions of the Line L through the point P0 (x0 , y0 , z0 ) and parallel to the vector v = ha, b, ci:
x = x0 + at,
y = y0 + bt,
z = z0 + ct
−
−
→
• Vector Equation of L: r = r0 + tv, where r0 = OP 0
• Scalar equation of plane through P0 (x0 , y0 , z0 ) with normal vector n = (a, b, c):
a(x − x0 ) + b(y − y0 ) + c(z − z0 ) = 0.
Often this will be written as a linear equation in x, y, z, ax + by + cz = d where d = ax0 + by0 + cz0 .
|ax1 + by1 + cz1 + d|
√
• Distance from a point P1 (x1 , y1 , z1 ) to the plane ax + by + cz + d is D =
a2 + b2 + c2
Examples
13. Find the vector equations of the line through the point P (3, 4, −1) and parallel to the y-axis.
14. Find the parametric and symmetric equation of the line through the points A(7, 1, −1) and B(−3, 3, 1).
15. Find an equation of
(a) the plane passing through the point A(1, 1, −9) and parallel to the plane 2y + z = 15.
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(b) the plane with x-intercept 5, y-intercept 4, and z-intercept 7.
(c) the plane that passes through the origin and contains the line x = 5t, y = −1 − t, z = −t.
16. Find the parametric equations of the line through the point P (8, 0, −10) and parallel to the line of
intersection of the planes 3x − y + 2z − 7 = 0 and x + 3y − 2z − 3 = 0.
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17. Verify that the given planes 2x − y − 3z + 4 = 0 and 4x − 2y − 6z + 12 = 0 are parallel and find the
distance between them.
18. Find the angle between the planes 2x + 2y − z = 4 and 6x − 3y + 2z = 5.
19. Find the point at which the line
x−4
z+6
=
, y = 8 intersects the plane x + y + z = 1.
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