(1) Identification
(a) General Equation of a plane in 𝑹𝟑 :
𝑨𝒙 + 𝑩𝒚 + 𝑪𝒛 = 𝑫 .
(b) Equations for Cone :
𝒛𝟐= 𝒙𝟐+ 𝒚𝟐 , 𝒚𝟐= 𝒙𝟐+ 𝒛𝟐 𝒐𝒓 𝒙𝟐= 𝒛𝟐+ 𝒚𝟐
(c) Equation of a Sphere and Inequality of a Ball
A sphere centered at (𝒂, 𝒃, 𝒄) with radis 𝒓 is the set of points satisfying the equation
(𝒙 − 𝒂)𝟐+ (𝒚 − 𝒃)𝟐+ (𝒛 − 𝒄)𝟐= 𝒓𝟐
A Ball centered at (𝒂, 𝒃, 𝒄) with radis 𝒓 is the set of points satisfying the inequality
(𝒙 − 𝒂)𝟐+ (𝒚 − 𝒃)𝟐+ (𝒛 − 𝒄)𝟐≤ 𝒓𝟐
Example 3 : What region in 𝑹𝟑 is represented by the following inequalities 𝟏 ≤ 𝒙𝟐+ 𝒚𝟐+ 𝒛𝟐≤ 𝟒 and 𝒛 ≤ 𝟎 ?
(2) Vector in 2-D
Vectors in Three Dimensions
(3) The Length of the three-dimensional vector 𝑎 =<
𝒂𝟏, 𝒂𝟐, 𝒂𝟑> is |𝒂| = √𝒂𝟏𝟐+ 𝒂𝟐𝟐+ 𝒂𝟑𝟐
(5) Cross Product
(6) Volume of a parallelepiped and a Tetrahedral
(b) Volume of a Tetrahedral
Volume of a Tetrahedral determined by the position vectors u, v,
and w is (1
6) |𝒖 ∙ 𝒗 × 𝒘|= 𝟏
𝟔 (that of the volume of the parallelepiped ).
(But why ?)
(a) Volume of a parallelepiped
Consider the parallelepiped (slanted box) determined by the position vectors u, v, and w (see figure). Show that the volume of the
parallelepiped is |𝒖 ∙ 𝒗 × 𝒘| = |
𝑢1 𝑢2 𝑢3 𝑣1 𝑣2 𝑣3 𝑤1 𝑤3 𝑤3
Solution :
(8) Equation of Lines in 3D
(a) Vector equations 𝒓 = 𝒓𝟎+ 𝒕𝒗
(b) Parametric equations
𝒙 = 𝒙𝟎+ 𝒂𝒕 , 𝒚 = 𝒚𝟎+ 𝒃𝒕, 𝒛 = 𝒛𝟎+ 𝒄𝒕 .
(c) Symmetric Equations
𝒙−𝒙𝟎
𝒂
=
𝒚−𝒚𝟎
𝒃
=
𝒛−𝒛𝟎
(9) Exercise
Identification
1) Find the distance between the pairs of points (- 1, -1, - 1) and (1, 1, 1).
2) Find the area of the triangle with vertices (1, 1, 0),
(1, 0, 1), and (0,1,1).
3) Describe (and sketch if possible) the set of Points in 𝑹𝟑 that
satisfy the given equation or inequality.
(a) 𝒙𝟐+ 𝒛𝟐 = 𝟒 ,
(b) 𝒛 ≥ √𝒙𝟐+ 𝒚𝟐 .
4) Describe (and sketch if possible) the set of Points in 𝑹𝟑 that
satisfy the given pair of equations or inequalities.
(a) {𝒙𝟐+ 𝒚𝟐+ 𝒛𝟐= 𝟒
𝒙𝟐+ 𝒛𝟐 = 𝟏
(b) {𝒙𝟐+ 𝒚𝟐= 𝟏 𝒛 = 𝒙
(c) {𝒙
𝟐+ 𝒚𝟐+ 𝒛𝟐≤ 𝟏
√𝒙𝟐+ 𝒚𝟐≤ 𝒛
5) Specify the boundary and the interior of the sets S in 3-space whose points (x, y, z) satisfy the given conditions. Is S open, closed, or neither?
𝑺 = {(𝒙, 𝒚, 𝒛): 𝟏 ≤ 𝒙𝟐+ 𝒚𝟐+ 𝒛𝟐≤ 𝟒} .
Vectors in 3D
6) 𝒖 = 𝟑𝒊 + 𝟒𝒋 − 𝟓𝒌 and 𝒗 = 𝟑𝒊 − 𝟒𝒋 − 𝟓𝒌. Find
(a) 𝒖 + 𝒗 , 𝒖 − 𝒗 , 𝟐𝒖 − 𝟑𝒗 ,
(b) the lengths |𝒖| and |𝒗| .
(c) unit vectors 𝒖̂ and 𝒗̂ in the directions of u and v, respectively,
(d) the dot product 𝒖 ∙ 𝒗 ,
(e) the angle between 𝒖 and 𝒗,
(f) the scalar projection of 𝒖 in the direction of 𝒗,
(g) the vector projection of 𝒗 along 𝒖.
7) Find the scalar and vector projections of vector 𝒃 =<
𝟓, −𝟏, 𝟒 > onto 𝒂 =< −𝟐, 𝟑, −𝟔 > .
8) Find the scalar and vector projections of vector 𝒂 = 𝒊 + 𝒋 + 𝒌 onto 𝒃 = 𝒊 − 𝒋 + 𝒌 .
9) 𝒂 =< 𝟏, 𝟏, −𝟏 > and 𝒃 =< 𝟐, 𝟒, 𝟔 >
Find the cross product 𝒂 × 𝒃 and verify that it is orthogonal to both 𝒂 and 𝒃.
Find the cross product 𝒂 × 𝒃 and verify that it is orthogonal to both 𝒂 and 𝒃.
Area of a Triangle
11) (a) Find a nonzero vector orthogonal to the plane through the points 𝑷(𝟏, 𝟎, 𝟏) , 𝑸(−𝟐, 𝟏, 𝟑) and
𝑹(𝟒, 𝟐, 𝟓) , and
(b) Find the area of triangle𝑷𝑸𝑹 .
12) (a) Find a nonzero vector orthogonal to the plane through the points 𝑷(𝟎, −𝟐, 𝟎) , 𝑸(𝟒, 𝟏, 𝟐) and
𝑹(𝟓, 𝟑𝟏, 𝟏) , and
(b) Find the area of triangle 𝑷𝑸𝑹 .
13) Area of a Parallelogram
Find the area of the parallelogram with vertices
𝑲(𝟏, 𝟐, 𝟑) , 𝑳(𝟏, 𝟑, 𝟔) , 𝑴(𝟑, 𝟖, 𝟔) 𝐚𝐧𝐝 𝑵(𝟑, 𝟕, 𝟑) .
Distance from a point 𝑷 to a Line 𝑳
14) (a) Let 𝑷 be a point not on the line 𝑳 that passes through the points 𝑸 and 𝑹. Show that the distance from the point 𝑷 to the line 𝑳 is
𝒅 =|𝒂 × 𝒃|
|𝒂|
where 𝒂 = 𝑸𝑹⃑⃑⃑⃑⃑⃑ and 𝒃 = 𝑸𝑷⃑⃑⃑⃑⃑⃑ .
(b) Use the formula in part (a) to find the distance from the point 𝑷(𝟏, 𝟏, 𝟏) to the line through 𝑸(𝟎, 𝟔, 𝟖) and
𝑹(−𝟏, 𝟒, 𝟕).
Distance from a point 𝑷 to a Plane 𝝅
15) (a) Let 𝑷 be a point not on the plane that passes through the points 𝑸, 𝑹 and 𝑺 . Show that the distance from 𝑷 to the plane is
𝒅 =|𝒂 ∙ (𝒃 × 𝒄)|
|𝒂 × 𝒃|
where 𝒂 = 𝑸𝑹⃑⃑⃑⃑⃑⃑ 𝒃 = 𝑸𝑺⃑⃑⃑⃑⃑ and 𝒄 = 𝑸𝑷⃑⃑⃑⃑⃑⃑ .
(b) Use the formula in part (a) to find the distance from the point 𝑷(𝟐, 𝟏, 𝟒) to the plane through the points
The volume of a Tetrahedron
16) A tetrahedron is a solid with four vertices 𝑷, 𝑸, 𝑹 𝐚𝐧𝐝 𝑺, and four triangular faces, as shown in the figure.
Let 𝒗𝟏, 𝒗𝟐, 𝒗𝟑 𝒂𝒏𝒅 𝒗𝟒 be vectors with lengths equal to the
areas of the faces opposite the vertices 𝑷, 𝑸, 𝑹 𝐚𝐧𝐝 𝑺
respectively, and directions perpendicular to the respective faces and pointing outward.
The volume of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face.
(a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices 𝑷, 𝑸, 𝑹 𝐚𝐧𝐝 𝑺 .
(b) Find the volume of the tetrahedron whose vertices
are 𝑷(𝟏, 𝟏, 𝟏), 𝑸(𝟏, 𝟐, 𝟑), 𝑹(𝟏, 𝟏, 𝟐) and 𝑺(𝟑, −𝟏, 𝟐) .
Equations of a straight Line in 3D
17) Find a vector equation and parametric equations for the line.
(a) The line through the point (𝟔, −𝟓, 𝟐) and parallel to the vector < 𝟏, 𝟑, −𝟐
𝟑>,
(b) The line through the point (𝟎, 𝟏𝟒, −𝟏𝟎) and parallel to the line
𝒙 = −𝟏 + 𝟐𝒕 , 𝒚 = 𝟔 − 𝟑𝒕 , 𝒛 = 𝟑 + 𝟗𝒕 .
18) Find parametric equations and symmetric equations for the line.
(a) The line through the origin and the point (4,3,-1) .
(b) The line through (𝟐, 𝟏, 𝟎) and perpendicular to both
𝒊 + 𝒋 and 𝒋 + 𝒌 .
19) Find a vector equation for the line segment from (𝟐, −𝟏, 𝟒)
to (𝟒, 𝟔, 𝟏).
20) Find parametric equations for the line segment from
(𝟏𝟎, 𝟑, 𝟏) to (𝟓, 𝟔, −𝟑) .
21) Find parametric equations for the line through the point
(𝟎, 𝟏, 𝟐) that is perpendicular to the line
𝒙 = 𝟏 + 𝒕 , 𝒚 = 𝟏 − 𝒕, 𝒛 = 𝟐𝒕
