Discovery guide 12.1 Discovery guide
Discovery 12.1.
(a)
Plot points \(P(1,2)\) and \(Q(3,-1)\) in the \(xy\)-plane. Draw an arrow from \(P\) to \(Q\text{.}\) This arrow is called the directed line segment \(\abray{PQ}\text{.}\)
(b)
Fill in the components for this directed line segment: \(\abray{PQ} = (\Delta x, \Delta y) = (\underline{\hspace{2.272727272727273em}},\underline{\hspace{2.272727272727273em}})\text{.}\)
(c)
On the same axes you've been working on, plot the point \(R\) that has the same coordinates as the components of \(\abray{PQ}\text{,}\) and draw the directed line segment \(\abray{OR}\) where \(O\) is the origin. What do you notice about this arrow?
(d)
This “common” arrow for \(\abray{PQ}\) and \(\abray{OR}\) (and all arrows in the \(xy\)-plane just like it) is called a vector. Let's label this vector \(\uvec{v}\text{.}\) Draw another “copy” of \(\uvec{v}\) so that its initial point is \(S(-2,1)\text{.}\) What will be the terminal point for this copy of \(\uvec{v}\text{?}\)
Discovery 12.2.
(a)
Draw the vector \(\uvec{u} = (2,3)\) with its initial point at \(P(1,1)\text{.}\) Label this vector on your diagram, and label its terminal point as \(Q\text{.}\) Now draw the vector \(\uvec{v} = (3,-1)\) with its initial point at \(Q\text{.}\) Label this vector on your diagram, and label this second terminal point as \(R\text{.}\) Draw in vector \(\uvec{w}\) corresponding to \(\abray{PR}\text{.}\)
(b)
Compute the components of \(\uvec{w}\) using a \((\Delta x, \Delta y)\) calculation between its initial and terminal points in your diagram, similarly to Task 12.1.b.
Looking at the components of \(\uvec{u}\) and \(\uvec{v}\text{,}\) what do you notice about the components of \(\uvec{w}\text{?}\) Based on this, we should call \(\uvec{w}\) the of \(\uvec{u}\) and \(\uvec{v}\text{,}\) and write \(\uvec{w} = \uvec{u}\underline{\hspace{0.909090909090909em}}\uvec{v}\text{.}\)
(c)
Now work in the reverse order: on the same diagram you've been working on, draw \(\uvec{v}\) starting at \(P\text{,}\) then draw \(\uvec{u}\) starting at that terminal point. Where did the second terminal point end up? Turn this into a rule for vector algebra: .
Your rule should be about the different ways to combine \(\uvec{u}\) and \(\uvec{v}\) that you've explored so far in this activity.
(d)
What shape do the four vectors on the outside make? And what is \(\abray{PR}\) relative to that shape (geometrically)?
Discovery 12.3.
(a)
How should you draw the vector \(\zerovec = (0,0)\text{?}\)
(b)
What happens if you draw \(\zerovec\) tail-to-head or head-to-tail with another vector (as in Discovery 12.2)? Turn this into a rule for vector algebra: .
Discovery 12.4.
We would reasonably expect \(\uvec{v} + (-\uvec{v}) = \zerovec\) in vector algebra.
(a)
Draw a geometric representation of this rule on a set of axes for \(\uvec{v} = (1,2)\) (use the origin as the initial point for \(\uvec{v}\)).
(b)
What should the components of \(-\uvec{v}\) be?
(c)
On the same set of axes as before, draw \(-\uvec{v}\) with its initial point at the origin.
Discovery 12.5.
Draw an arbitrary vector in the plane, and label it \(\uvec{u}\text{.}\) Then draw another arbitrary vector with its initial point at the terminal point of \(\uvec{u}\) (but maybe have this new vector head off in a new direction). Label this second vector \(\uvec{v}\text{.}\) Now draw in the sum vector \(\uvec{w} = \uvec{u} + \uvec{v}\text{,}\) similarly to Discovery 12.2.
(a)
Which of your three vectors represents \(\uvec{w} - \uvec{u}\text{?}\)
(b)
Draw in another vector for \(\uvec{u} - \uvec{w}\text{.}\)
(c)
What is the point of this activity?
Discovery 12.6.
(a)
Draw a representative diagram for the vector sum \(\uvec{v} + \uvec{v}\) using \(\uvec{v} = (1,2)\) (start with the first initial point at the origin). What are the components of this sum vector?
From both the geometry of what you've drawn, and the result for the components of the sum vector \(\uvec{v} + \uvec{v}\text{,}\) do you think it is reasonable to write \(2 \uvec{v}\) to mean \(\uvec{v} + \uvec{v}\text{?}\)
(b)
Now draw each of the following, and determine their components: \(3 \uvec{v}\text{,}\) \(-2 \uvec{v}\text{,}\) \(\frac{1}{2} \uvec{v}\text{,}\) \(-\frac{5}{4} \uvec{v}\text{.}\)
Discovery 12.7.
Draw an arbitrary representative diagram for \(\uvec{w} = \uvec{u} + \uvec{v}\) (similarly to how you started Discovery 12.5). On the same set of axes, draw a diagram for \(2 \uvec{u} + 2 \uvec{v}\text{,}\) and compare with \(2 \uvec{w}\text{.}\) Express what you've discovered as a rule of vector algebra, with \(2\) replaced by variable \(k\text{:}\) .
Discovery 12.8.
On a set of \(xy\)-axes, draw the standard basis vectors: \(\uvec{e}_1 = (1,0)\) and \(\uvec{e}_2 = (0,1)\text{,}\) along with the vector \(\uvec{v} = (5,2)\text{.}\) Then draw a geometric representation of \(\uvec{v}\) as a linear combination \(\uvec{v} = 5 \uvec{e}_1 + 2 \uvec{e}_2\text{.}\)
Discovery 12.9.
All the vectors we've encountered so far are two-dimensional vectors. Let's bump everything up a dimension.
(a)
Using \(\uvec{u} = (1,1,0)\) and \(\uvec{v} = (1,-1,2)\text{,}\) draw the following on a set of \(xyz\)-axes: \(\uvec{u}\text{,}\) \(\uvec{v}\text{,}\) \(\uvec{u} + \uvec{v}\text{,}\) \(-\uvec{u}\text{,}\) \(\uvec{v} - \uvec{u}\text{,}\) \(2 \uvec{v}\text{.}\)
(b)
Now compute the components of each of the vectors from the previous part of this activity.
We can't draw pictures of \(n\)-dimensional vectors if \(n>3\text{,}\) but we can do all the same algebra.
Discovery 12.10.
Complete the following vector algebra formulas.
- \(\displaystyle (u_1,u_2,\dotsc,u_n) + (v_1,v_2,\dotsc,v_n) = \underline{\hspace{6.818181818181818em}}\)
- \(\displaystyle (u_1,u_2,\dotsc,u_n) - (v_1,v_2,\dotsc,v_n) = \underline{\hspace{6.818181818181818em}}\)
- \(\displaystyle -(v_1,v_2,\dotsc,v_n) = \underline{\hspace{6.818181818181818em}}\)
- \(\displaystyle k(v_1,v_2,\dotsc,v_n) = \underline{\hspace{6.818181818181818em}}\)
- \(\displaystyle \uvec{0} = \underline{\hspace{6.818181818181818em}}\)
We can use vectors to represent other kinds of “displacements” besides position displacements. Vectors can be used to represent change between states of any collection of related variables.
Discovery 12.11.
An investor sinks $10,000 into stock in each of companies A, B, C, and D. After a year, the various items in her portfolio have the following values: company A, $10,475; company B, $11,240; company C, $9,756; company D, $10,054.
Represent the “displacement” in the collection of values of the investor's holdings, from initial state
to terminal state
as a four-dimensional vector.
Discovery 12.12.
If we write two-dimensional vectors in the form \(\uvec{u} = \left[\begin{smallmatrix} u_1 \\ u_2 \end{smallmatrix}\right]\text{,}\) instead of the form \(\uvec{u} = (u_1,u_2)\text{,}\) then we can use matrix algebra to do computations with vectors.
(a)
Does each rule of vector algebra that we've discovered today have a counterpart rule in matrix algebra?
(b)
Will the same be true for the algebra of higher-dimensional vectors? (That is, if we consider using \(n\times 1\) column matrices to represent \(n\)-dimensional vectors?)