Section 12.2: Vectors

3D Review

Questions

What is the distance formula for the distance between two points in space, ((Subscript[x, 0]),(Subscript[y, 0]),(Subscript[z, 0])) and ((Subscript[x, 1]),(Subscript[y, 1]),(Subscript[z, 1]))?

Find the distance between (1,2,3) and (-4,4,2).

Questions

Consider the set of points (x,y,z) that are a distance of 2 from the origin (0,0,0).

What is an equation that x,y,z must satisfy for (x,y,z) to be in this set?

What is this shape?

Generalize this to find an equation for any of this shape.

Questions

The equation (((x-2))^2)+(y^2)+(((z+3))^2)=4 represents a sphere.

What is its center?

What is its radius?

Describe its intersections with each of the coordinate planes.

Questions

Identify the set.

y=-2

The equation (x^2)+(y^2)+(z^2)+2x-4y-10z=0 represents a sphere.

What is its center?

What is its radius?

z>4

Vectors

Definition

A vector is an object with direction and magnitude.

Example

While driving at one particular instance my speedometer shows that I’m going 53 mph and my gps shows I’m headed 30° north of east. My velocity vector has magnitude (a speed -- 53 mph) and direction (30° north of east). This is a 2D vector.

I’m standing at a particular point on a mountaintop. I need to tell a helicopter pilot how to reach me. I give her a direction and a distance to fly to reach me. The position vector from the helicopter to me has  magnitude (the distance to fly to me) and the direction. This is a 3D vector.

I need to move my refrigerator. To overcome friction I need to pull with a magnitude of 30 pounds and it needs to be away from the back wall in a perpendicular fashion. I need to pull with a force vector that has magnitude (30 pounds) and direction (directly away from the wall).

Vector graphics

Represent a vector with an arrow whose length is the magnitude of the vector and it points in the direction of the vector

Graphics3D[{Arrow[{{0,1,2},{2,1,3}}]},Boxed->True,Axes->True]

Questions

Let ((Subscript[x, 0]),(Subscript[y, 0])) be the initial point of the arrow and ((Subscript[x, 1]),(Subscript[y, 1])) be the terminal point of the arrow where the arrowhead is.

Draw a rectangle with the vector as one of the diagonals.

What are the dimensions of the rectangle?

What is the length of the arrow?

Find the direction of the vector as an angle with the positive x-direction.

Vector components

The vector components for a vector are the dimensions of the rectangle if the vector is 2D, dimensions of the box if the vector is 3D.

Questions

What are the components of the vector whose initial point (1,2) and terminal point (3,1)?

What is the magnitude of this vector?

Find the direction of this vector as an angle with the positive x-axis.

What are the components of the vector whose initial point (4,2,-3) and terminal point (1,2,3)?

What is the magnitude of this vector?

Notation

A vector is written in component form in two different ways. If vector v has x-component a, y-component b, and z-component c:

(Overscript[u, ->])=⟨a,b,c⟩

(Overscript[u, ->])=a (Overscript[i, ->])+b (Overscript[j, ->])+c (Overscript[k, ->])

The vectors (Overscript[i, ->]),(Overscript[j, ->]), and (Overscript[k, ->]) are called unit vectors, because they have unit length. They are also mutually perpendicular, and so they comprise what is called an orthogonal basis for 3-space. That’s a mouthful. They’re orthogonal because they’re mutually perpendicular; they’re a basis because one can write any vector in 3-space in terms of them.

Questions

Draw vector ⟨1,-2⟩ with its initial point at (0,0).

Draw vector 2(Overscript[i, ->])+3(Overscript[j, ->]) with its initial point at (-1,1).

The vectors (Overscript[i, ->]) and (Overscript[j, ->]) are an orthogonal basis for 2-space.

Vector arithmetic

Arithmetic is done on vectors only when it has geometric significance.

Scalar multiplication

A scalar is just a single number as opposed to a vector that is multiple information packaged in one object. Scalar multiplication:
	scalar * vector=vector

Questions

Let λ be the scalar 2 and (Overscript[v, ->])=⟨1,2⟩. What should the components of 2(Overscript[v, ->]) be?

Draw both (Overscript[v, ->]) and 2(Overscript[v, ->]) with the same initial point. How are the two related geometrically?

Draw both (Overscript[v, ->]) and -(1/2)(Overscript[v, ->]) with the same initial point. How are the two related geometrically?

Questions

In general, how should λ⟨a,b,c⟩ be computed?

What is the geometric significance of λ (Overscript[v, ->]) as compared to (Overscript[v, ->]).

Given the three vectors (Subscript[(Overscript[v, ->]), 1])=⟨4,12,-8⟩,(Subscript[(Overscript[v, ->]), 2])=⟨-3,-9,6⟩,(Subscript[(Overscript[v, ->]), 3])=⟨2,2,-6⟩, are any parallel to each other?

Vector addition

Component form

If (Overscript[u, ->])=⟨a,b,c⟩ and (Overscript[v, ->])=⟨e,f,g⟩, then (Overscript[u, ->])+(Overscript[v, ->])=⟨a+e,b+f,c+g⟩

Geometric significance

Equivalent forms:

Draw vector (Overscript[u, ->]). Place the initial point of vector (Overscript[v, ->]) at the terminal point of (Overscript[u, ->]). Draw the vector sum with its initial point at the initial point of (Overscript[u, ->]) and its terminal point at the terminal point of (Overscript[v, ->]).

Clear[u, v]; uu = {2, 1}; vv = {1, -1};
Graphics[{Arrow[{{0, 0}, uu}], Arrow[{uu, uu + vv}], Arrow[{{0, 0}, uu + vv}], Text[u, .5 uu, {1, -1}], Text[v, uu + .5 vv, {-1, -1}], Text[u + v, .5 (uu + vv), {0, 1}]}, BaseStyle -> {FontSize -> 18}]

Draw vectors (Overscript[u, ->]) and (Overscript[v, ->]) with initial points together. The two vectors form two sides of a parallelogram. Complete the parallelogram. Draw the vector sum from the initial points of  (Overscript[u, ->]) and (Overscript[v, ->]) to the opposing vertex of the parallelogram.

Clear[u,v];uu={2,1};vv={1,-1};
Graphics[{Arrow[{{0,0},uu}],Arrow[{{0,0},vv}],Arrow[{{0,0},uu+vv}],Dashing[{0.02}],Line[{uu,uu+vv,vv}],Text[u,.5uu,{1,-1}],Text[v,.5vv,{-1,-1}],Text[u+v,.5(uu+vv),{0,1}]},BaseStyle->{FontSize->18}]

Questions

Let (Overscript[u, ->])=⟨1,1⟩ and (Overscript[v, ->])=⟨1,2⟩.

Draw (Overscript[u, ->])+(Overscript[v, ->]) as in the first geometric significance.

Draw (Overscript[u, ->])+(Overscript[v, ->]) as in the second geometric significance.

Questions

Using the same two vectors, what is the geometric significance of (Overscript[u, ->])-(Overscript[v, ->])? Draw it.

Questions

Consider the triangle with vertices A((Subscript[a, 1]),(Subscript[a, 2])),B((Subscript[b, 1]),(Subscript[b, 2])),C((Subscript[c, 1]),(Subscript[c, 2])).

Graphics[{Line[{{0,0},{1,2},{2,-.5},{0,0}}],Text[A,{-.1,-.1}],Text[B,{1,2+.1}],Text[C,{2.1,-.6}]},BaseStyle->{Large}]

What is vector (Overscript[A B, ->]) in component form? What is vector (Overscript[B C, ->]) in component form?

What is the vector sum (Overscript[A B, ->])+(Overscript[B C, ->]) in component form?

Geometrically what is the vector sum (Overscript[A B, ->])+(Overscript[B C, ->])?

Question

Why is it true that the line joining the midpoints of two sides of a triangle must be parallel to the third side?

Question

There are the two component notations for vectors, (Overscript[u, ->])=⟨a,b,c⟩ and (Overscript[u, ->])=a (Overscript[i, ->])+b (Overscript[j, ->])+c (Overscript[k, ->]). Expressions (Overscript[i, ->]),(Overscript[j, ->]),(Overscript[k, ->]) are vectors themselves. What are they?

Unit vectors

Definition

A unit vector is a vector with magnitude 1. Its magnitude is fixed, but its direction can be any direction.

Questions

Let (Overscript[u, ->])=⟨1,2⟩

Is (Overscript[u, ->]) a unit vector?

Find a scalar λ such that λ (Overscript[u, ->]) is a unit vector.

Find a general method for taking a vector and finding a unit vector pointing in the same direction.

Does this work for all vectors?

Questions

What are the unit vectors parallel to the tangent line to the curve y=2sin(x) at the point (π/6,1)?

What are the unit vectors perpendicular to the tangent line to the curve y=2sin(x) at x=π/6?