12.1. Cartesian Space
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1 12.1. Cartesian Space In most of your previous math classes, we worked with functions on the xy-plane only meaning we were working only in 2D. Now we will be working in space, or rather 3D. Now we will have 3 axes, x, y, and z. Figure We will now have 8 octants instead of 4 quadrants. Figure
2 Example 1: Plot the point ( 1, 3, 2). Example 2: Give the equation of a plane that is parallel to the xz-plane that passes through the point ( 1, 3, 2). 2
3 Important Formulas: (1) Distance formula: d(p 1, P 2 ) = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 + (z 2 z 1 ) 2 ( x1 + x 2 (2) Midpoint formula:, y 1 + y 2, z ) 1 + z (3) Equation of a sphere: (x a) 2 + (y b) 2 + (z c) 2 = r 2 Figure Example 3: Give the equation of the sphere that has A(2, 0, 1) and B(2, 1, 3) as the endpoints of a diameter. 3
4 12.3. Vectors A vector is an ordered triple (in space) where addition and multiplication by scalars holds. Vectors have a direction and a length (magnitude or norm). Figure Example 1: If the point P has coordinates of (0, 2, 3) and the point Q has coordinates (2, 1, 4), find the vector P Q. The commutative and associative properties of addition hold for vectors: (1) Commutative property: a + b = b + a (2) Associative property: (a + b) + c = a + (b + c) The zero vector is: 0 = (0, 0, 0). Vectors can be multiplied by a scalar value. Example 2: Set a = ( 5, 2, 0) and b = (1, 3, 2) and find 2a 3b. 4
5 Two vectors are parallel if a = αb for some real number α. If α > 0, then a and b have the same direction. If α < 0, then a and b have opposite directions. Example 3: Determine if any of the following vectors are parallel: a = (1, 1, 2), b = (2, 1, 2), c = ( 3, 3, 6) The norm (or magnitude) of a vector a = (a 1, a 2, a 3 ) is found by a = a a2 2 + a2 3 Unit Vectors are vectors of norm 1. To find a unit vector, we divide the vector by its norm: u a = a a There are three special unit vectors:. i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) Figure All vectors can be represented by a linear combination of these: (a 1, a 2, a 3 ) = a 1 i + a 2 j + a 3 k 5
6 12.4. The Dot Product The dot product of two vectors a = (a 1, a 2, a 3 ) and b = (b 1, b 2, b 3 ) is found by a b = (a 1 )(b 1 ) + (a 2 )(b 2 ) + (a 3 )(b 3 ) Notice that the dot product of two vectors is commutative. Also note the fact that the dot product gives a scalar answer. The dot product is used in many formulas: (1) Cosine of the angle between two vectors: cos θ = a b a b (2) The component of a in the b direction: comp b a = a u b (3) The projection of a in the b direction: proj b a = (a u b )u b Figure Example 1: Givent a = i 2j + 3k and b = 2i + j + k find the angle between a and b as well as the component and projection of a in the b direction. 6
7 Example 2: Find all numbers x for which the angle beetween (x, 1, 1) and (1, x, 1) is π 3. The angles α, β, γ that a vector makes with unit vectors i, j, and k are called direction angles of a. A unit vector with these direction angles is: cos αi + cos βj + cos γk. Figure
8 12.5. The Cross Product The cross product of two vectors a = (a 1, a 2, a 3 ) and b = (b 1, b 2, b 3 ) is found by a b = (a 2 b 3 a 3 b 2 )i + (a 3 b 1 a 1 b 3 )j + (a 1 b 2 a 2 b 1 )k Notice that the cross product of two vectors is NOT commutative (in fact, it is anticommutative). Also note the fact that the cross product gives a vector answer. The cross product is really the determinant of a 3x3 matrix: i j k a b = a 1 a 2 a 3. b 1 b 2 b 3 Example 1: Verify the formula for the cross product using the formula for determinant. 8
9 Example 2: Calculate (2i j + 2k) (i 3k). If vectors a and b are NOT parallel, they form two adjacent sides of a parallelogram: Figure The vector a b is perpendicular to the plane containing this parallelogram. If a b then a b = 0. The area of the parallelogram formed by vectors a and b is A = a b. Example 3: Find the area of triangle P QR given P (2, 0, 3), Q(3, 1, 0), R(2, 3, 2). 9
10 If vectors a, b and c are noncoplanar, then they form sides of a parallelepiped. The volume of a parallelepiped is found by V = (a b) c. Note that (a b) c is called the scalar triple product of vectors a, b and c. Figure Example 4: Find the volume of the parallelepiped who sides are the vectors a = (2, 0, 1), b = (1, 1, 2), and c = ( 1, 1, 2). 10
11 12.6. Lines The vector equation r(t) = r 0 + td (t is a real number) parameterizes the line l. The tip of r 0 is the point (x 0, y 0, z 0 ) and d (the direction vector) is (d 1, d 2, d 3 ) so we can write l as: r(t) = (x 0 + t d 1 )i + (y 0 + t d 2 )j + (z 0 + t d 3 )k The form of the line given above is in vector form. We can also write lines in scalar parametric form and symmetric form. Scalar parametric form: x(t) = x 0 + td 1, y(t) = y 0 + td 2, z(t) = z 0 + td 3 Symmetric form: x x 0 d 1 = y y 0 d 2 = z z 0 d 3 Example 1: Find a vector parameterization for the line that passes through the point P (1, 2, 0) and is parallel to the line r(t) = (i j + 2k) + t(i + k) Example 2: Re-write the answer to example 1 in scalar parametric form then in symmetric form. 11
12 Two lines are parallel if their direction vectors are scalar multiples of each other and they have no points in common. If they have direction vectors that are scalar multiples of each other and they have points in common, we say the lines coincide. If two lines l 1 : r(t) = r 0 + td and l 2 : R(u) = R 0 + ud intersect then there exists numbers t and u such that r(t) = R(u). Lines that are not parallel, coincident or intersecting are called skew. Example 3: Determine if the following lines are parallel, coincident, intersecting or skew: l 1 : r(t) = (3i + j + 5k) + t(i j + 2k) l 2 : R(u) = (i + 4j + 2k) + u(j + k) 12
13 Other important formulas involving lines l 1 : r(t) = r 0 + td and l 2 : R(u) = R 0 + ud: (1) Cosine of the angle between two lines: cos θ = u d u D (2) The distance from point P to line l 1 : d(p, l 1 ) = P r 0 d d 13
14 12.7. Planes We have worked with planes that are perpendicular to the x, y or z-axis such as the plane z = 2 graphed below: Figure What about planes that are not perpendicular to an axis? Let N = Ai + Bj + Ck be the nonzero vector perpendicular to a plane through point P (x 0, y 0, z 0 ) and let Q(x, y, z) be another point on the plane as shown: Figure Notice that the vector P Q = (x x0, y y 0, z z 0 ) and because N is perpendicular to our plane, N P Q = 0. Using this information, we find the equation of the plane to be: (A, B, C) (x x 0, y y 0, z z 0 ) = 0 A(x x 0 ) + B(y y 0 ) + C(z z 0 ) = 0 14
15 Example 1: Find an equation for the plane which passes through the point P (1, 2, 3) and is perpendicular to the vector i j + 2k Example 2: Find an equation in x, y, z for the plane that passes through the given points. P 1 (3, 4, 1), P 2 (3, 2, 1), P 3 (1, 1, 2) 15
16 Given two planes P 1 and P 2, then exactly one of the following holds: (1) P 1 and P 2 are parallel. (2) P 1 and P 2 coincide. (3) P 1 and P 2 intersect in a straight line. If the planes intersect, the direction vector of their line of intersection is found by the cross product of their normal vectors. Example 3: Given P 1 : x + y + z + 1 = 0 and P 2 : x y + z + 2 = 0, determine if the planes are parallel, coincident or intersect. If they intersect, determine the line of intersection. Other important formulas involving planes: (1) Cosine of the angle between two lines: cos θ = u N1 u N2 (2) The distance from point P (x 1, y 1, z 1 ) to plane P : Ax + By + Cz + D = 0: d(p, P) = Ax 1 + By 1 + Cz 1 + D A2 + B 2 + C 2 16
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