Chapter 12 Vectors and the Geometry of Space

Given a vector a, write it in terms of it's direction cosines

(pg. 850)

Theorem 9: state and prove. How to find the magnitude of a cross product?

(pg. 857)

The cross products various combinations of unit vectors

(pg. 858)

What is the scalar triple product of the vectors a, b, and c? Express it using a determinant

(pg. 859)

What is a cone?

(pg. 877)

What is a hyperboloid of one sheet?

(pg. 877)

Symmetric equations of a line given a point on the line (x₀, y₀, z₀) and the direction numbers <a,b,c>

(x - x₀)/a = (y - y₀)/b = (z - z₀)/c (pg. 865)

Properties of the cross product

1) Shows that the cross product is NOT commutative. 2) shows that it doesn't matter where we multiply by a scalar. 3) and 4) are the left and right distributive properties. 5) Is the SCALAR triple product 6) is the VECTOR triple product (pg. 859)

Properties of vectors

1) commutative 2) associative 3) any vector plus the zero vector is itself 4) The sum of any vector with the negative of itself is the zero vector 5) scalar distribution 6) scalar distribution 7) ... 8) any vector times 1 is itself (pg. 842)

List the five properties of the dot product.

1) tells us the dot product of a vector with itself is equal to the square of its length. 2) tells us the dot product is commutative 3) distributive 4) it doesn't matter when we multiply by a scalar 5) The dot product of any vector with the zero vector is zero (pg. 847)

What information do we need to specify a plane in R^3?

A point in the plane and a vector orthogonal to the plane is all that information we need to specify a plane (pg. 867)

What is a hyperbolic paraboloid?

A quadric surface that looks like a saddle (pg. 876)

What is an ellipsoid?

A quadric surface whose traces are all ellipses (think of a squashed sphere) (pg. 875)

What is an elliptic paraboloid?

A quadric surface whose traces are ellipses and parabolas (pg. 876)

What is a vector?

A quantity that has both magnitude and direction (think of an arrow) (pg. 838)

What is a cylinder?

A surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve (pg. 874)

What is the normal vector of a plane?

A vector that is orthogonal to the plane. In other words, its a vector that's orthogonal to any vector in the plane. A normal vector is found by taking the cross product of two linearly independent vectors in the plane (pg. 867)

What is a position vector of a point P?

A vector whose initial point is the origin and whose terminal point is P (pg. 840)

What is a unit vector?

A vector whose length is one. A unit vector that point in the direction of some nonzero vector P is P divided by the length of P (pg. 843)

The length/magnitude of two and three dimensional vectors

An easier equation to remember is the square root of the dot product of the vector with itself (pg. 841)

How to calculate the vector and scalar projections of b onto a

Both of these are derived from theorem 3: a.b=|a|*|b|*cosθ (pg. 851)

Given the x and y axes, how is direction of the z axis determined?

By the right hand rule. The direction of z is given by xXy (pg. 832)

Derive the vector equation of a plane (equations 5 and 6)

Given a point in the plane P(x₀, y₀, z₀) and a normal vector n, we reason that the normal vector must be orthogonal to any vector in the plane, namely (r-r₀). So we have n . (r - r₀) = 0 which is equivalent to n . r = n . r₀ (pg. 867)

Derive the parametric equations for a line

Given a point on the line P (x₀, y₀, z₀) and a vector V = <a, b, c> in the direction of the line, parametric equations for the line are x = x₀ + at, y = y₀ + bt, z = z₀ + ct (where t is free) or <x, y, z> = <x₀ + at, y₀ + bt, z₀ + ct> (where t is free) These are derived by breaking the vector equation of the line into three scalar equations. (pg. 864)

What are the direction numbers of a line? Why are they useful?

Given a vector in the direction of a line, the components of that vector v = <a,b,c> are called the direction numbers of the line. The direction numbers are the three dimensional analogue to the slope of a line (m) in R^2. (pg. 865)

Define the initial and terminal points of a vector.

Given a vector that starts at a point A and ends at a point B, then A is the initial point and B is the initial point (pg. 838)

Definition of work

Given force and displacement vectors, the work is their dot product (pg. 852)

Derive the scalar equation of a plane (equation 7)

Given normal vector n = <a,b,c>, r₀ = <x₀, y₀, z₀>, and r = <x, y, z>, we substitute these into the vector equation of a plane to obtain the scalar equation of the plane (pg. 867)

Definition of the dot product for three dimensional vectors

Given two vectors in R^n, and b, their dot product is given by finding the sum of the products of their corresponding components (pg. 847) aka scalar product and inner product

What are the components of a vector?

If a vector is written as v=<x,y,z> then the components of v are x,y, and z; geometrically, they tell us how far we have to move along each axis to reach the terminal point of the vector (pg. 840)

How can we tell if two lines are parallel?

If their direction numbers are proportional

Given vectors a and b, how can we classify the angle between them?

If their dot product is positive, then θ is acute If their dot product is zero, then θ=π/2 If their dot product is negative, then θ is obtuse. If a and b point in exactly opposite directions, then we have θ=π and their dot product is equal to the negative product of their magnitudes. Everything on this card comes directly from theorem 3 (pg. 850).

How do we know if two planes are parallel?

If their two normal vectors are parallel (pg. 868)

Derive the symmetric equations of a line

If we have the parametric equations for the line, we can solve each equation for t and equate the results, thus eliminating the parameter. The result is called the symmetric equations of the line (pg. 865)

Theorem 3: State and Prove

If θ is the angle between vectors a and b, then their dot product is equal to the product of their magnitudes times cosθ (pg. 848) We prove this using the law of cosines

What is the length and direction of the zero vector?

It has zero length and has no direction (pg. 838)

What is the geometric interpretation of the dot product?

It's given in terms of the angle between vectors and and b. Given vectors a and b, their dot product is the product of their lengths times the cosine of the angle between them where 0≤θ≤π. This allows us to predict when vectors are orthogonal to each other: orthogonal vectors have a zero dot product (pg. 848)

Given two vectors, a and b, what is the vector projection of b onto a? How is it calculated?

It's the component of b that lies in the direction of a (pg. 851)

Explain what O is.

It's the origin. The point (0, 0, 0) (pg. 832)

Given that two planes intersect, how do we find the (acute) angle between them?

It's the same angle as the angle between their normal vectors (pg. 868)

Given two vectors, a and b, what is the scalar projection of b onto a? How is it calculated?

It's the signed magnitude of the vector projection of b onto a. It's calculated by using the equation a.b=|a|*|b|*cosθ and solving for |b|*cosθ (pg. 851)

Given a point P with coordinates (a,b,c), what are the projections of P onto the coordinate planes?

Let Q be the projection of P onto the xy-plane, S be the projection of P onto the xz-plane, and R be the projection of P onto the yz-plane. Then Q has coordinates (a,b,0), S has coordinates (a,0,c), and R has coordinates (0,b,c) (pg. 833).

Given two vectors a and b in R^3, derive their cross product

Let c be some vector that is orthogonal to both a and b; equations 1 and 2 come from dotting c with a and b respectively. Eliminating c3 produces equation 3, from which c1 and c2 can be produced using inspection. Substituting these values into 1 and 2 yield c3 (pg. 854)

What are skew lines?

Line that are not parallel and that do not intersect (pg. 866)

Are the direction numbers of a line unique?

No, they are not unique since any vector in the direction of the line can be used in defining that line. Given a set of direction numbers <a,b,c>, the other possible direction numbers are given by t<a,b,c> (where t is free) (pg. 865)

Derive the vector equation of a line L

Notice that all we need is a point on the line and a vector parallel to the line. The line is traced out by the vector r as t ranges from negative infinity to positive infinity (pg. 863)

Definition of the cross product

Notice that the cross product, unlike the dot product is a vector rather than a scalar. The cross product of two vectors is always orthogonal to both vectors and its direction is given by the right hand rule (pg. 855)

Definition of the standard basis vectors. Why do we use these?

Notice these all have a length of 1 and point along the standard x,y,z axes. We use the standard basis vectors because they form a basis for Vn (pg. 842)

What are octants? Where is the first octant?

Octants are the 3D analogue to quadrants. The first octant is determined by the positive coordinate axes; in other words, the first octant is the set of all points whose coordinates are all positive (pg. 832)

What are the direction cosines of a vector a?

The cosines of the direction angles: cosα, cosβ, cosγ. We can find these using corollary 6 (see picture) The direction cosines of a are the components of the unit vector in the direction of a (pg. 850)

Theorem 8: The vector axb is orthogonal/parallel to both a and b? Prove it

The cross product of any two vectors is orthogonal to those vectors (pg. 856)

What are the direction angles?

The direction angles of a nonzero vector a are the angles α, β, γ (in the interval [0, π]) that a makes with the positive x, y, and z axes respectively (pg. 85)

Find the symmetric equations of a line given two points on that line (x₀, y₀, z₀) and (x₁, y₁, z₁). Also explain how to find the direction numbers

The direction numbers are given by a = x₁ - x₀, b = y₁ - y₀, c = z₁ - z₀. The symmetric equations are then (x - x₀)/(x₁ - x₀) = (y - y₀)/(y₁ - y₀) = (z - z₀)/(z₁ - z₀) (pg. 866)

What is a quadric surface?

The graph of a second-degree equation in three variables x,y, and z (pg. 875)

What is the geometric interpretation of the cross product?

The length/magnitude of the cross product axb is equal to the area of the parallelogram determined by a and b (pg. 858)

Given the linear equations of two planes in R^3, how can you tell if they are perpendicular?

The linear equations will be in the following form a1x + b1y + c1z + d1 = 0 a2x + b2y +c2z + d2 = 0 The normal vectors are then v1 = <a1, b1, c1> v2 = <a2, b2, c2> If the normal vector are perpendicular, the planes are perpendicular, so take their dot product and see if it's zero

Length/magnitude of an n-dimensional vector

The square root of the dot product of the vector with itself

What is the geometric significance of the scalar triple product of vectors a, b and c?

The volume of the parallelepiped determined by the vectors a, b, and c is the length/magnitude of their scalar triple product (pg. 859-860)

What are the coordinate axes?

The x,y, and z axes in R^3 (pg. 832)

What are the coordinate planes?

The xy, yz, and xz planes (pg. 832)

What's the cross product of any vector in V3 with itself?

The zero vector (pg. 856)

2) Parametric equations for a line given a point on the line and a vector in the direction of the line

These are derived from the vector equation of the line (pg. 864)

Explain what the triangle and parallelogram laws are

They show us how to add vectors. The parallelogram law shows that vector addition is commutative. (pg. 838)

Corollary 6

This follows immediately from theorem 3 (pg. 849)

Derive the vector equation for a line segment from the tip of vector r₀ to the tip of vector r₁

This is derived by taking the vector equation of the line through the tips of those vectors, r = r₀ + tv and replacing v with r₁ - r₀. The restriction 0 ≤ t ≤ 1 gives the line segment (pg. 866)

Cauchy-Schwartz Inequality

This is easily proven using theorem 3 and the fact that |cosθ|≤1 (exercise 61 pg. 854)

T/F: The scalar projection of b onto a is the same thing as the scalar projection of a onto b

This is false (pg. 851)

T/F: axb = bxa

This is false: axb and bxa point in opposite directions and so are different vectors (they will have the same length though) (pg. 859)

T/F: the vector equation and parametric equations of a line are unique

This is false: we can arbitrarily choose any point on the line and any vector parallel to the line to construct the equations (pg. 864)

T/F: ax(bxc) = (axb)xc

This is false; the associative law for cross products does NOT hold. http://mathforum.org/kb/message.jspa?messageID=5761160

Explain how to add and subtract vectors given their components. Explain how to scale a vector given its components

To add vectors, we add their components. To subtract vectors, we subtract their components. We scale a vector by multiplying all the components by the scalar (pg. 841)

Define torque in terms of the position and force vectors

Torque is the cross product of the position and force vectors. Notice that torque is a vector. Torque measures the tendency of a body to rotate about the origin. The direction of the torque indicates the axis of rotation (pg. 860)

T/F any two vectors with the same length and direction are equal, regardless of their positions

True. The positions of the vectors does not matter, only that they have the same length and direction (pg. 838)

How do we find the direction of axb?

Using the right hand rule (pg. 856)

What do the symbols V2, V3, and Vn represent?

V2 is the set of all two dimensional vectors. V3 is the set of all three dimensional vectors. Vn is the set of all n-dimensional vectors (pg. 842)

Vector equation of line L

We just need a point on the line (x₀, y₀, z₀) from which we construct the position vector r₀ = <x₀, y₀, z₀> and a direction vector v = <a, b, c> which is analogous to the slope of the line in two dimensional Cartesian space. The vector r = r₀ + tv (t is free) will give us every point on the line (pg. 864)

How do we show that three vectors are coplanar?

We show that their scalar triple product is zero (pg. 860)

How do we represent a point in space?

With an ordered triplet (a,b,c) (pg. 832)

Is vector addition commutative?

Yes, a+b=b+a. See the parallelogram law (pg. 839)

Given three vectors, a, b, and c, what is their vector triple product?

ax(bxc) (pg. 860)

Derive the distance between a point P₁(x₁, y₁, z₁) to a plane ax + by + cz + d = 0.

example 8 (pg. 869)

Distance formula in three dimensions; prove it.

pg. 835

Equation of a sphere with center C(h,k,l) and radius r

pg. 835

Explain how to add vectors

pg. 838

Definition of scalar multiplication.

pg. 839

Given points A(x1,y1,z1) and B(x2,y2,z2), explain how to find the vector AB

pg. 840

Show why the associative law of vector addition is true

pg. 842

Cross product in terms of a determinant

pg. 855

linear equation of a plane in R^3

pg. 867

redo problem 377 frrom 12.55

pg. 872

What is a parabolic cylinder?

pg. 874

List the types and equations of six different quadric surfaces

pg. 877

What is a hyperboloid of two sheets?

pg. 877

Theorem 7: two vectors are orthogonal iff

their dot product is 0 (pg. 849)

Explain how to subtract vectors

u-v=u+(-v) we just add negative one times the vector we want to subtract (pg. 839)

Theorem 10: If two vectors are parallel or anti-parallel iff their cross product is ...

zero because the sin0=sinπ=0 (pg. 857)