Calculus 3: Vectors

√[( x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]

3D distance formula

addition of vectors

Draw vectors tip-to-tail.

(x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2 (where x0, y0, and z0 are center values)

Equation of a Sphere (hollow)

midpoint

M (upper case)

linear combination

Name the form: -2i + 5j

standard position (a position vector)

if a vector has its initial point at the origin, it is in ___

parallel

if θ= 0 or 180 the two vectors are ___

perpendicular

if θ= 90 the two vectors are ___

0 or 180

if θ= ___ the two vectors are parallel

90

if θ= ___ the two vectors are perpendicular

slope

m (lower case)

collinear points

points that lie on the same line

parallel

same directions means...?

same magnitude

same length means...?

magnitude

the length of a vector

position vector (tail at origin)

v = <v1, v2> What kind of vector does the above describe?

Ratios are equivalent

vectors are parallel if

parallel vectors

vectors with the same or opposite direction

triangle rule

when two vectors are drawn to scale in tip-to-tail fashion, the vector drawn from the tail of the first vector to the tip of the second vector is the resultant of these two vectors

Coinitial Vectors (since they have the same initial point 'A'.)

→AB and →AC are examples of what kinds of vectors?

point (represents a dot) vector (represents a vector)

(3,-5) is in ___ notation, whereas <3,-5> is in ___ notation

(x-x0)^2 + (y-y0)^2 + (z-z0)^2 <= r^2 (where x0, y0, and z0 are center values)

Equation of a ball (solid)

Find unit vector, scale by l ( If v is the given vector: l*<a/|v|, b/|v|> result will be a vector. )

Find the following vector. The vector in the direction of <a,b> with length l (How do you solve this?)

its magnitude (don't forget to simplify and rationalize)

To find the unit vector, divide each component of a vector by ___

Identity vectors

^i = <1,0> ^j = <0,1> ^k = <0,0,1>

zero vector

a vector with both the initial and terminal point at the origin (pretty much just a dot)

position vector

a vector with its tail at the origin and its head at the point (v1, v2). It is written as <v1, v2>, and is said to be in standard position.

True

(T or F) All perpendicular vectors are independent, but not all independent vectors are perpendicular. BUT all independent vectors are non-parallel

(x1+x2/2, y1+y2/2, z1+z2/2) (add the corresponding x components of each point and divide the result by 2. Repeat this for the y and z components)

3D midpoint formula

parametric equations

A line that passes through a point P(x0, y0, z0) and is parallel to vector v= <a, b, c> is represented by

Vector

A quantity that has magnitude and direction

directed line segment

A segment between two points A and B with a specified direction, from A to B or from B to A

<15,9> <-15,-9> (Parallel just means going in same direction. Scaling a vector does not change its direction. To get a vector with 3x the magnitude of the given vector, just scale the vector by 3. Take care that you don't confuse this with finding a parallel vector with a magnitude of 3. That would require you to make a unit vector from v and then scale it by 3.)

Find two vectors parallel to v with three times the magnitude of v. v = <5,4>

Compare magnitudes (If we are asking how far these points are from the origin, then we can write them as position vectors with the same component values since we would be subtracting (0,0,0). Then, we simply calculate the magnitudes of our new vectors and compare. The greater the magnitude, the further from the origin.)

If given two points and asked to determine which is farther from the origin, what should you do?

parallel

If the cross product of two vectors is the zero vector, 0i + 0j + 0k, the magnitude of the cross product is 0, and the vectors are...?

orthogonal

If the dot product of two vectors is 0, the vectors are...?

parallel

If two vectors are ___, their dot product will be equal to the product of their respective magnitudes.

the zero vector (mag of 0)

If two vectors are linearly dependent, then their cross product is ___

parallel

If two vectors are linearly dependent, then they are ___

not parallel (not necessarily orthogonal either though)

If two vectors are linearly independent, then they are ___

1. Equation of a ball. 2. The center point of the ball and the radius. 3. Completing the square. (CtS quick summary: get like terms together get constant values on right side of = mark. take half of coefficient value of first degree term, then square it. take those values, add them to the right side of = mark. take square root of constant (right side of = mark) for radius. take post halved, pre squared values from coefficients, flip signs, those are center point values. )

If you are given the following: x^2 + y^2 + z^2 - 4x - 10y - 12z <= 74 1. What are you looking at? 2. What can you find? 3. How do you find it/them?

12√5 (Although the normal vectors created by changing the order of the cross product are going in opposite directions, their magnitude (length) is equal. For further proof, just consider that the magnitude formula requires each component to be squared which eliminates any negatives.)

If |u x v| = 12√5, |v x u| will = ?

1. Their dot product is = to 0 2. <-5,9,7> o <3,a,b> = -15 + 9a + 7b 3. Infinitely many 4. Set -15 + 9a + 7b equal to zero and solve for one variable: -15 + 9a + 7b = 0 9a + 7b = 15 7b = 15 - 9a b = (15 - 9a)/7 5. <-15, a, (15 - 9a)/7> 6. All reals.

Let a and b be real numbers. Find all vectors <3,a,b> orthogonal to <−5,9,7> 1. How do you determine whether two vectors are orthogonal? 2. Perform operation 3. How many vectors are orthogonal to <-5,9,7> (or any other vector)? 4. Knowing the answer to #3, what should you do now? 5. Present your answer in terms of one variable. 6. Knowing the answer to #3, what values might that variable have to make the vector orthogonal to <-5,9,7>?

square root of the sum of components squared √(u₁^2 + u₂^2)

Magnitude of a vector

0

Orthogonal (perpendicular or normal) vectors have a dot product of ___.

perpendicular

Orthogonal, or normal, means

a and c are the same length and are going in the same direction b and d are the same length, but they are going in different directions

See image. Explain why vectors a and c are equal, but b and d are not.

1. a and c 2. b c and d (a and c are equal because they have the same length and are going in the same direction b, c, and d are coinitial because they all originate at the same spot )

See image. Identify: 1. Equal vectors 2. Coinitial vectors

The direction vector

See image. Where do the a, b, and c values come from?

The point values (The "anchor")

See image. Where do the x0, y0, and z0 values come from?

origin

Standard position vector, position vector, and component form all refer to a vector whose tail is at the ___

cross product

The ___ of two vectors, u and v, is orthogonal to both u and v.

direction cosines of a vector

The angles between the axes and the vector.

unit vector

The identity vectors are also what type of vector?

scalar

The result of a dot product will be a ___

all other vectors

The zero vector 0 is parallel to ___

Terminal - Initial

To find the component form of a vector with two points, use the formula:

5<a,b> - 6<5,1> = <3,-5> Distribute: <5a,5b> + <-30,-6> = <3,-5> Add: <5a - 30, 5b - 6> = <3,-5> Set each component = to its desired value and solve: 5a - 30 = 3 and 5b - 6 = -5 5a = 33 a = 33/5 5b = 1 b = 1/5 Declare solution: x = <a,b> x = < 33/5, 1/5 >

Use the properties of vectors to solve the following equation for the unknown vector. x=<a,b> Let u=<5,1> and v=<3,−5>. Use: 5x−6u=v (Just describe the process of solving the problem)

+ or - |u|*|v| (product of magnitudes)

Vectors u and v are parallel iff u dot v = ?

z = -2

What is the equation of the plane parallel to the xy​-plane through (5​,3​,-2​)?

y = 9

What is the equation of the plane parallel to the xz-plane through (6​,9​,8​)?

<5,7,4> (Position vector = component form = terminal - initial)

What position vector is equal to the vector from ​(−8​,1​,5​) to ​(−3​,8​,9​)?

zero 0 any ()

When the initial and terminal points of a vector are the same point, then the vector is a ___ vector with a magnitude of ___ and is going in ___ direction.

2i + 7j

Write the following vector as a linear combination <2,7>

line

Two different planes intersect in a ___

collinear

Two vectors are ___ if they lie on the same line or parallel lines.

equivalent vectors

Two vectors are equal if they are parallel and of equal length. (we called them equivalent above) Therefore, if we translate (move) a vector it is still considered the same vector, (the location does not matter only direction and length).

<6,3,-2> = 7< 6/7, 3/7, -2/7> (Find |v|, make the unit vector, put the magnitude of v out in front of the unit vector)

Write the vector v = <6, 3, −2> as a product of its magnitude and a unit vector with the same direction as v.

|uxv| (magnitude of the cross product (determinant) )

area of a parallelogram having u and v as adjacent sides

|uxv|/2 (magnitude of the cross product (determinant) divided by two)

area of a triangle having u and v as adjacent sides

unit vector

A vector with a magnitude (length) of 1

zero vectors (Each capital letter represents a point. The first letter is the initial point, and the second letter is the terminal point. in AA and BB, both the initial and terminal points are the same point. Which means it starts and stops in the exact same spot.)

AA, BB, etc. are examples of what kind of vector?

negative (BA is the same vector as AB, just going in the opposite direction)

BA is the ___ of vector AB

cross product (determinant)

How do you find the normal (orthogonal) vector of two vectors?

Standard unit vectors Coordinate unit vectors

Identity vectors are also known as...

Colinear

If two vectors are parallel (going in same direction) and share a common point, then they are ___

The product of the magnitudes of the vectors

If two vectors are parallel, their dot product is = to...?

the product of the magnitudes of the vectors.

If two vectors are perpendicular, the magnitude of the cross product of the vectors is = to...?

< 20, -16, -8 > (When you find a cross product you are solving for the determinant of a 3x3 matrix. u x v and v x u are the exact same matrices except for one key difference: the 2nd and 3rd row are interchanged because of the order of u and v being swapped. If you'll remember, interchanging two rows has the effect of flipping the sign of the determinant. Therefore, if we know the determinant of one (u x v), we can find the determinant of the other (v x u) simply by swapping the signs of the components. Note that this also has the effect of turning the normal vector around, so u x v and v x u are going in opposite directions, but are still parallel.)

If u x v is < -20, 16, 8 >, what is v x u?

turns it around (reverses its direction)

Multiplying a vector by a negative value does what to the vector?

makes it shorter

Multiplying a vector by a value between 0 and 1 (proper fraction) does what to the vector?

Makes it longer

Multiplying a vector by a value greater than 1 does what to the vector?

component form (initial centered at origin)

Name the form: < 2, -1 >

orthogonal

The cross product of two vectors, u and v, is ___ to both u and v.

component

To find the magnitude of a vector, it must be in ___ form

parallelogram rule

To find the resultant of two non-parallel vectors, construct a parallelogram wherein the two vectors are adjacent sides. The diagonal of the parallelogram shows the resultant.

orthogonal vectors

two vectors whose dot product equals zero

initial terminal

All vectors have an ___ point and a ___ point.

cos(theta) = dot product of the 2 vectors / the product of the magnitudes of the two vectors u o v / |u|*|v|

Angle between two vectors (Find the angle)

equal vectors

two vectors that have the same magnitude and the same direction

2. Opposite directions 2. Parallel

u x v and v x u are going in ___ and are ___ to each other. 1. The same direction 2. Opposite directions 3. Similar directions 1. Perpendicular 2. Parallel

perpendicular parallel

u ⋅ v = 0 means ___ | u x v | = 0 means ___

2 (Create two vectors, ->AB and ->AC, (other combos are valid but for simplicity's sake we'll just make two co-initial vectors) then test them for parallelism using either ratio comparison, dot product, or cross product)

Given 3 points, A, B, and C, and asked to determine whether the points are collinear, it is fairly obvious that vectors will need to be created and checked for parallelism. The question is, how many vectors must be created?

A = xy B = xz C = yz P = (4,2,4) (A, B and C are all 2D planes that have one coordinate = to 0. To find the coordinates of P, start at the origin, and count how many units P is in the x direction. Then go back to the origin and count to y. repeat for z. )

Based on the image, determine which planes A, B, and C represent, and give the coordinate values of point P.

1 points

The 3D distance and midpoint formulas are used with ___ 1. Points 2. Vectors

direction (vector) points (anchors)

When writing the equations for lines in space (3D) and planes, there are an infinite number of possible equations because although the ___ will be the same, there are an infinite number of possible ___ you could use, and each point will give you a different equation.

Coinitial Vectors

two or more vectors which have the same initial point. ex. →AB and →AC