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- University of Delaware
- Mathematics
- Mathematics 243
- Hague
- Final Exam, Chapter 12

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**a • b** = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}
**a • b** = |**a**| |**b**| cos(**ᶿ**)
**a** x **b** = < a_{2}b_{3} - a_{3}b_{2}, a_{3}b_{1} - a_{1}b_{3}, a_{1}b_{2} - a_{1}b_{1} >
**a** x **b** = 0
**a • **( **b **x **c **)
**r** = **r**_{0} +t**v**, where**r** is the *position vector***r** is the *original position vector***t** is the *parameter***v** is the *given vector*
**r**_{0} = < x_{0} , y_{0}, z_{0} > **v** = < a, b, c >

_{0} + at y = y_{0} + bt z = z_{0} + ct
__x - x___{0} = __y - y___{0} = __z - z___{0}

**r**(t) = ( 1 - *t* ) **r**_{0 }+ *t***r** _{1} ; where ( 0 ≤ *t *≤ 1 )
^{z}/_{c} = x^{2}/a^{2}** + ** y^{2}/b^{2}

^{z}/_{c} = x^{2}/a^{2} **-** y^{2}/b^{2}

**- **x^{2}/a^{2} **-** y^{2}/b^{2} + z^{2}/c^{2} = 1

(12.3) The Dot Product, **a • b**

(12.3) If ᶿ is the angle between the vectors **a** and **b **...

(12.3) If ᶿ is the angle between the nonzero vectors **a** and **b **...

cos(ᶿ) = **a • b** / |**a**| |**b**|

(12.3) What makes two vectors orthogonal?

Two vectors are orthongal is **a • b** = 0

(12.3) Scalar Projection of **b** onto **a**

comp_{a}**b** = **a • b** / |**a**|

(12.3) Vector Projection of **b** into **a**

proj_{a}b = ( **a • b** / |**a**|^{2} ) **a**

(12.3) The Displacement Vector

(12.4) The Cross Product, **a** x **b**

(12.4) If ᶿ is the angle between the vectors **a** and **b**, and ( 0 ≤ ᶿ ≤ ∏ )...

|**a **x **b**| = |**a**| |**b**| sin(**ᶿ**)

(12.4) What makes two nonzero vectors **a **and **b** parallel?

(12.4) Scalar Triple Product form

(12.5) Vector Equation of 3d form

(12.5) Parametric Equations of a vector **L** through a point *P*

NOTE: **r** = < x, y, z >

< x,y,z > = < x_{0} + at, y = y_{0} + bt, z_{0} + ct >

x = x
(12.5) Symmetric Equations of the Vector **L** through a point *P*

a b c

We eliminate the *t *parameter from the parametric functions

(12.5) The Vector Equation for the line segment from **r**_{0} to **r** _{1}

(12.5) Skew lines

Lines that do not intersect with and are not parallel to one another

(12.5) Parallel Vectors

Vectors are parallel when they are scalar multiples of the same unit vector

(12.5) Vector Equation of a Plane through a point *P* with normal vector **n**

NOTE:**n ** = < a, b, c >**r **** **= < x, y, z >**r**_{0}** = < x**_{0}** , y**_{0}**, z**_{0}** > **

a ( x - x_{0}) + b ( y - y_{0}) + c ( z - z_{0}) = 0

(12.5) The distance **D** between a plane and a point on a vector

(12.6) Quadric surfaces, *Ellipsoids*

x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2} = 1

All traces are ellipses

It a = b = c, the ellipsoid is a sphere

Picture a football!

(12.6) Quadric surfaces, *Elliptic Paraboloids*

Horz. traces are ellipses

Vert, traces are parabolas

The variable raised to the first power indicates the axis of the paraboloid

Picture half a football!

(12.6) Quadric surfaces, *Hyperbolic Paraboloids*

Horz. traces are hyperbolas

Vert. traces are parabolas

(12.6) Quadric surfaces, *Cones*

z^{2}/c^{2} = x^{2}/a^{2}** +** y^{2}/b^{2}

Horz. traces are ellipses

Vert. traces in the planes *x = k* and *y = k* are:

hyperbolas when *k ≠ 0*

a pair of lines when *k = 0*

Picture an hourglass

(12.6) Quadric surfaces, *Hyperboloid of One Sheet*

x^{2}/a^{2} + y^{2}/b^{2} **-** z^{2}/c^{2} = 1

Horz. traces are ellipses

Vert. traces are hyperbolas

The axis of symmetry corresponds to the variable whose coefficient if negative

Picture an hourglass with a wide neck

(12.6) Quadric surfaces, *Hyperboloid of Two Sheets*

Horz. traces in z = k are ellipses if *k > c* or *k < -c*

Vert. traces are hyperbolas

The two minus signs are indicative of the two sheets

Picture two domes not touching and opposite each other

The cross product and parallelograms

The length of the cross product **a** x **b** is equal to the area of the parallelogram determined by **a** and **b**

I.E. Find lengths of two vectors, then find the cross product using these values

About this deck

Author: Ethan M.

Created: 2011-05-19

Updated: 2011-07-20

Size: 26 flashcards

Views: 23

Created: 2011-05-19

Updated: 2011-07-20

Size: 26 flashcards

Views: 23

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