1229 Math Midterm 1

What are the components of vector x?

(x,y,z)

What is the formula for the normal vector n when given three points P, Q, R?

n = (Q-P) \times (R-P) (you can choose any of the three points for P)

What does the normal vector n represent?

perpendicular to a surface or plane

How are the components of the normal vector n related to the coefficients of the standard form equation?

n = coefficients of standard form when the plane is perpendicular

What is the formula for the normal vector n when a plane is parallel to two vectors u and v?

n = (u \times v)

What is a unit vector?

A vector with a length (magnitude) of 1

Under what condition are two values or vectors orthogonal?

value that is orthogonal: when dot product rule = 0

vector that is orthogonal: when cross product = 0

When is the cross product of two vectors 0?

when the vectors are orthogonal

When are vectors considered collinear?

when they have the same 0 components or their ratios are the same

What is the standard form or linear equation of a plane?

ax+by+cz=d

In standard form (ax+by+cz=d), what do the coefficients a, b, c represent?

They are the coefficients of the normal vector n

How do you calculate standard form using a formula

n \cdot x = n \cdot p

What is the normal point form of the equation of a plane?

n \cdot (x-p)

What is the formula for the volume of a parallelepiped defined by vectors u, v, w?

|(u \times v) \cdot w|

What is the formula for the area of a parallelogram defined by vectors u and v?

length of (u \times v)

What is the formula for the area of a triangle, and how is it derived from a parallelogram?

1/2 of the area of the parallelogram. If given 3 points, convert them to 2 vectors by using the equivalent formula.

What is the formula for the dot product involving the angle between two vectors, and its rearrangement for \cos\theta?

(u \cdot v) = (||u||)(||v||)(\cos\theta) \longrightarrow rearranged formula for \cos\theta

Describe the Right-Hand Rule for the cross product u \times v.

pointer = u
m-iddle = v
-humb = u \times v

What are the key properties of a cross product?

for R^3 only,
-ot commutative (can’t switch order),
-nswer is a vector

What angle defines two vectors as perpendicular?

90^\circ

How is the dot product of a vector with itself (u \cdot u) related to its length?

(u \cdot u) is the same as (||u||)^2

What is a formatting rule for dot products or vectors written in i, j, k form?

no commas for dot product or i,j,k form

How do the coordinates of endpoints relate to the components of a vector?

the coordinates of endpoints is the same as the components of the vector

What is the notation convention for points versus vectors?

points = capital case
-ectors = lower case

How is the distance between two vectors determined?

the distance of two vectors is the distance of their endpoints

What strategy should be used when points are given instead of vectors for writing equations?

use equivalent formula

What does t represent in parametric equations?

parameter

What does R^m denote?

m-dimensional Euclidean space

How are 'planes' described in R^m?

For R^m have dimension m-1 and instead of -lane it’s called Hyperplane

What variable notation is used for coordinates in R^m?

instead of doing x,y,z,d,e etc use x1,x2,x3,x4,x_5 etc

What type of vector is required for point normal form and standard form equations?

need a vector that's perpendicular

How do you find a vector perpendicular to a 2D vector?

just do the same coordinate but swap order and make one neg

Geometrically, when are two vectors collinear (passing through the origin)?

when line -adding through origin is same line as both vectors

Describe the effect of scalar multiplication (c) on a vector's direction and length.

if c>0 then new line is same direction but longer , if c<0 then new line is opposite direction and longer

What is the formula for the distance between a point Q and a plane (using standard and point normal form)?

Plug Q into x or x,y,z then keep numerator the same and put the length of n as denominator. Numerator is absolute. (\text{Distance} = \frac{|axQ+byQ+cz_Q-d|}{||n||})

Describe the point parallel form equation for a line and its vector requirement.

x(t) = P + tV \longrightarrow (needs a vector that is parallel not perpendicular) use this formula to get a point on R^3 by plugging in any number into t

How are parametric equations written for a line in 3D space?

same formula x(t) = P + tV but now also for y and z (plug in points in order)

What is the two-point form for the equation of a line?

x(t) = (1-t)P+tQ

What is the formula for distance?

square root: (p1-q1)² + (p2-q2)²

What is the formula for length?

square root: (p1)² + (p2)²

What is the formula for the equivalent vector?

(q1-p1 , q2-p2 , q3-p3)

What is the formula for a non zero vector in same direction?

(1/length of vector)(vector)

What is the formula for a non zero vector in opposite direction?

(-1/length of vector)(vector)

Directed line segments (vectors) always start at which coordinate?

(0,0) origin

Either than the basic equivalent formula, what are two other ways you can think of it as?

As:

(endpoint of line segment) - (start point)

or

V = Q - P