Math 2433 Notes – Week 1 Session 1 12.1 Cartesian Space Coordinates

Ex: Give the equation of a plane that is parallel to the xz-plane that passes through the point (2, -1, 4).

Distance Formula: 2 2 2

1 2 2 1 2 1 2 1( , ) ( ) ( ) ( )d P P x x y y z z= − + − + −

Midpoint Formula: 1 2 1 2 1 2, ,2 2 2

x x y y z z+ + +⎛ ⎞⎜ ⎟⎝ ⎠

Equation of a Sphere: ( ) ( ) ( )2 2 2 2x a y b z c r− + − + − = Examples: 1) Give the equation of the sphere that has A and B as the endpoints of a diameter. A (2, 1, 0) B (1, 1, -3) 2) Find the center and radius of 2 2 2 4 8 2 5 0x y z x y z+ + + − − + =

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).

Properties of vectors: Commutative: a + b = b + a Associative: (a + b) + c = a + (b + c) The zero vector 0 = (0,0,0) (note: a = ⋅0 0) Vectors can be multiplied by a scalar: if a = ( )1 2 3, ,a a a , then 2a = 2a1,2a2 ,2a3( ) The norm of a vector a = ( )1 2 3, ,a a a is 2 2 2

1 2 3a a a= + +a

Examples: 1) Find the vector PQ

and determine its norm given points P and Q.

P(5,3,2), Q(-3,1,5)

2) Set a = (-5, -2, 6), b = (3, 0, 4), c = (-5, 1, 5). Find: 4a + b - 3c 3) Simplify the linear combination:

4(2j - 3k) + 2(2i + 3j - 4k)

Two vectors are parallel if a = α b 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. 4) Are any of the following vectors parallel?

a = (1, -1, 2) b = (2, -1, 2) c = (3, -3, 6) d = (-2, 2, -4) Unit Vectors are vectors of norm 1.

a =aua

au has direction a

5) Find the unit vector for a = (3,4,-2) There are 3 special unit vectors: i = (1,0,0) j = (0,1,0) k = (0,0,1) All vectors can be represented by a linear combination of these: ( )1 2 3, ,a a a = 1a i + 2a j + 3a k Why? 6) Calculate the norm of the vector:

7i + 3j - 4k

7) Find α given 3i + j – k and αi – 4j + 4k are parallel 8) Find α so that the norm of αi +( α-1)j + (α+1)k is 2. 9) Find the vector of norm 2 in the opposite direction of a = i + 2j - k

10) Let a = (7, 5, 2), b = (6, 4, 1), c = (7, 5, 7), and d = (4, 4, 6). Find scalars A, B, C such that d = Aa + Bb + Cc

12.4 The Dot Product Given a = (a1, a2, a3) and b = (b1, b2, b3), a i b = (a1)(b1) + (a2)(b2) + (a3)(b3) Note that this gives an answer that is NOT a vector. The dot product gives an answer that is a scalar. Examples: 1) a = (-1, -3, 5), b = (2, 3, -4) a i b = 2) a = 6i + 5j + 4k, b = i + 3j a i b =

The angle between two vectors is found with this formula: cosθ = a i b

a b

3) Given, a = 4i + 4j, b = i - 2j + 3k, c = 2i + 2k Find the angle between a and b Find the angle between a and c

Projection of a on b projba = (a i ub )ub where a i ub = compba

4) Given a = 4i + 3j, b = i - 3j + 2k, c = 2i + 4k Find the projection of a in the b direction. Find the component of a in the c direction.

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: cosα i + cosβ j+ cosγ k

5) Find a unit vector with direction angles: α = π⁄6, β = π⁄3, γ = 3π⁄2 6) Find the direction angles of a = i - 3k

7) Find all numbers x such that

2i + 4j + 2xk ⊥ 6i + 3j - 4xk

12.5 Cross Product If vectors a and b are NOT parallel, they form two sides of a parallelogram: a b a ×b is the vector perpendicular to this plane. So, if a || b then a ×b = 0 The area of this parallelogram is A= || a ×b || for a = ( )1 2 3, ,a a a and b = ( )1 2 3, ,b b b , a ×b = Example 1) Calculate (3i + j + k) × (i – 2j)

2) Find two unit vectors perpendicular to a = (1,2,-1) and b = (1,0,2) 3) Find the area of the parallelogram with vertices:

A(2, 1, 4), B(1, 4, 3), C(1, 0, 2), D(2, -3, 3)

Volume of a Parallelepiped: V= |(a × b) i c |

4) Find the volume of the parallelepiped with the given edges.

i - 3j + k, 3j - k, i + j - 3k