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In previous lessons, we've discussed geometric objects and ideas as largely static things. We've calculated distance, volume, area, angles, and equations at given moments in time. Even if these objects are moving, we have only dealt with them at one instant, performing calculations which disregard all information except the properties that describe them at one particular instant. While these kinds of calculations are useful, we have not yet been able to capture quantities that rely on more than one descriptor. For example, we can describe an object's speed with a single number, but to describe its velocity, we also need to know the direction it's moving in. We call speed a scalar and velocity a vector for this reason.
Vectors are very useful when describing real-world phenomena. It can be hard to visualize complex three dimensional equations, and vectors are one way mathematicians use to make it easier for themselves. They enable us to understand geometry in three-dimensional space, as well as to describe direction and displacement of points, and determine angles and lengths in situations that are too complicated to visualize.
To begin our exploration of vectors, let's consider how we represent movement in two dimensions.
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Discussion
Say you want to get from point A to point B in the diagram below.
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If you consider a single "step" to be moving horizontally or vertically by one number, how many steps do you need to take
to the right?
up?
As a shorthand, let's call each step to the right one "i-step" and each step up one "j-step." How many i- and j- steps do you need to take to get from the start to the end? Can you express your movement as a sum of steps in both directions?
We can visualize the steps you take in each direction as the base and height of a triangle whose hypotenuse is most direct line from the start to the end, drawing arrows at the end of the lines to denote that these aren't just lines, but movement with a direction.
If you wanted to get from point A to point Q, how many i- and j- steps would you need to take, as a sum?
How many of each kind of step would you need if you wanted to go from Q to B, as a sum?
How about from P to B?
Putting all of these ideas together, can you use the same kind of expression to describe moving
one step right, one step up
one step left, two steps right
zero steps horizontally, three steps down
in terms of i- and j-steps?
Part (a)
From A(0,0) to B(3,2), the change in each coordinate tells us how many steps of each kind we need:
So we take three i-steps (to the right) and two j-steps (up). As a sum of individual steps:
Or more compactly,
Part (b)
From A(0,0) to Q(1.5,1):
From Q(1.5,1) to B(3,2):
From P(3,0) to B(3,2):
Part (c)
For any move, write the displacement as Δ=ai+bj where a is the net i-steps and b the net j-steps.
– One step right, one step up
– One step left, two steps right
– Zero steps horizontally, three steps down
The notion of splitting movement into several components that describe multiple directions is captured cleanly by vector notation. Vectors are labelled using either a bold lower case letter (e.g. v) or a lower case letter with an arrow "hat" on top (e.g. v⃗). (Note that the two notations are equivalent.)
Vectors are usually drawn as directed line segments. Any vector can be uniquely by a diagram showing its length and direction, or numerically by giving its horizontal and vertical components. These are two numbers describing the number of units in each direction required to get from the "tail" to the "head" of the arrow.
As you just saw, we often describe these components by giving i- and j- "steps."
2D vector notation with base vectors
The equation of a vector in two dimensions can be written as a sum of its vertical and horizontal components, denoted i and j respectively. These are vectors that start from the origin and point right (i) and up (j), each with a length of 1. We call these base vectors.
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Checkpoint
The diagram below depicts a vector v.
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Write v as a sum of base vectors.
Select the correct option
Another way to represent the information encoded in a vector is using column vector notation.
2D vector notation with column vectors
A vector in two dimensions can be uniquely described by giving its vertical and horizontal components, meaning the number of units in each direction required to get from the "tail" to the "head" of the arrow. These components can be represented using a column vector.
The column vector of a vector v whose head is located a horizontal and b vertical units from its tail is
The top number represents the horizontal component and the bottom number represents the vertical component.
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Checkpoint
The diagram below depicts a vector v.
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Write v as a column vector.
Select the correct option
It's important to note that these notations are equivalent. Both are frequently used. When giving your answers on IB exams, you can use whichever notation you prefer.
Checkpoint
A vector is defined v=3i−j.
Write v as a column vector.
Select the correct option
Thus far, we've discussed vectors as two dimensional objects lying on the Cartesian plane. But the greatest utility of vectors is in describing the real world, which is not a flat sheet.
We can extend our basic definitions to a third dimension easily by adding a third component, k, the describes movement and direction along the z-axis. Just as i and j are perpendicular to each other, k is perpendicular to both of these two. Think of k as the direction coming directly towards you, out of your computer screen.
3D base and column vectors
A vector is a mathematical object that has both magnitude (size) and direction in space. A three-dimensional vector is typically described by its components along the x, y, and z axes.
In 3 dimensions, a vector is expressed as
where i, j, and k are called the base vectors:
The point P(a,b,c) can be described by a vector starting from the origin, where v=⎝⎛abc⎠⎞.
Checkpoint
The position of a point on the three-dimensional grid is given by the coordinates P=(−4,0,2).
Write down the description of the vector v⃗ ending at P. Give your answer in the form v⃗=v1i+v2j+v3k.
Select the correct option
Discussion
We've already used addition to describe vectors: describing v⃗ by adding its i, j, and k components. You might wonder if we can extend this concept to the addition of different vectors v1⃗ and v2⃗.
Let v1⃗ and v2⃗ be vectors given by
How would you describe the sum v1⃗+v2⃗?
A vector is completely determined by how far it moves you along each mutually perpendicular axis. Writing
“doing” v⃗1 then v⃗2 means you first move x1 in the i-direction, then x2 in the same direction, so in total you move x1+x2 along i. The same holds independently in the j- and k-directions. By distributivity and commutativity,
Geometrically, you’re just adding each independent “component” of displacement.
In our case
so
or in i,j,k form
This means the combined displacement is 7 units along i, 1 unit along j, and 1 unit along k.
Adding & subtracting vectors
To add two vectors, add their corresponding components:
To subtract two vectors, subtract their corresponding components:
Addition corresponds to placing vectors head-to-tail, and subtraction corresponds to the displacement between their tips.
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Example
The diagrams below show the sum v1⃗+v2⃗ and difference v1⃗−v2⃗ changing as you change the values of v1⃗ and v2⃗.
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Checkpoint
Two vectors are given by v=⎝⎛56−7⎠⎞,w=⎝⎛3−42⎠⎞.
Find the sum v+w.
Select the correct option
So far, we've mostly worked with vectors that start at the origin. But we can also discuss displacement vectors, which are those that go from one point on the coordinate grid to another.
Another way of writing vectors is to label the ends points with capital letters and put an arrow "hat" on top of the
Discussion
In the diagram below, the purple vector stretches from point A to point B. The vectors starting at the origin with A and B as their end points are drawn as blue and red dashed lines respectively.
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Can you describe the vectors that go from the origin to the points A and B as column vectors?
Based off of this, can you write the purple vector as a column vector?
How would you describe the purple vector in words?
Think of the purple arrow as “the step you need to take at A to land at B.” Here’s one way to see why it comes from subtracting two origin‐arrows:
Pick some origin O and draw the dashed blue arrow from O to A. Call that arrow OA. Then draw the dashed red arrow from O to B; call it OB. These two arrows just show where A and B sit in space.
Now ask: “How do I get from A back to O, and then on to B?”
– Going from A to O is exactly the reverse of OA, so it’s “– OA.”
– Then you follow OB.
Altogether that move is
If you slide this combined arrow so its tail starts at A, its tip reaches B—that’s exactly the purple arrow AB. In symbols:
Finally, in coordinates, if
then those dashed arrows are
In words: it tells you “go (b1−a1)” units in the x–direction, “go (b2−a2)” in the y–direction, and “go (b3−a3)” in the z–direction, starting from A to arrive exactly at B.
We call the vectors from the origin to A and B position vectors, and the purple vector from A to B a displacement vector.
Position & Displacement Vectors
A position vector describes the position of a point relative to a fixed origin. For example, the position vector of a point P describes the location of P relative to an origin O:
A displacement vector describes how to move from one point to another. For instance, the displacement vector from point A to point B is given by subtracting position vectors:
Thus, position vectors identify points relative to an origin, while displacement vectors represent movement or translation between points. Expressing AB as OB−OA is sometimes referred to as giving the relative position of B from A.
Example
If OB=i−2j+3k and OA=2i+j+k, find AB.
Checkpoint
Point A has coordinates (0,4,−3). Point B has coordinates (2,−21,5).
Find the displacement vector from point A to point B.
Select the correct option
Exercise
Given that OP⃗=⎝⎛−60−3⎠⎞ and PQ⃗=⎝⎛131⎠⎞, find the position vector of point M which lies halfway between points P and Q.
Select the correct option
Just as vectors can be added and subtracted from one another, they can also be manipulated via multiplication. The simplest kinds of multiplication we can apply to a vector are by zero and negative one.
Zero vector and negative vector
The zero vector 0 is a special vector without size or a defined direction, represented by
The negative of a vector reverses the vector's direction while maintaining its size. If v=⎝⎛xyz⎠⎞, then the negative is
pointing in exactly the opposite direction.
Checkpoint
The vector v is defined v=⎝⎛5k−2⎠⎞.
Find the value of k that makes the expression
true.
Select the correct option
Now that we know vector addition and multiplication, you might wonder about vector multiplication and division. In fact we've already learned how to multiply by zero and by negative one. These numbers are called scalars, meaning they are a single value (1,4,√2,π, etc.) instead of a list of multiple like vectors are. Scalars are what you've spent most of your mathematical life working with up to this point.
Discussion
Remember that the negative of a vector reverses its direction:
Based on what we already know about the negative of a vector, how do you think we could extrapolate this process to multiplying v⃗ by any number k?
To see how to multiply any scalar k by a vector v⃗=(x,y,z)T, start with the two easiest cases:
In each case we multiply every component by the scalar. By exactly the same process, for any real number k we set
So multiplying a vector by k simply stretches or shrinks (and possibly reverses, if k<0) each component by the factor k.
This process is called scaling a vector.
Scalar multiples of vectors
Scaling a vector means multiplying it by a number k, which changes its length without affecting its direction (unless k<0, which reverses direction). If
then scaling by k gives
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Checkpoint
The vector v=⎝⎛32−5⎠⎞ is multiplied by some scalar k.
Find the value of k such that
Select the correct option
Exercise
Given u=⎝⎛13−2⎠⎞ and v=⎝⎛−60−3⎠⎞, find 3u−32v.
Select the correct option
If one vector is a scalar multiple of another, we call them parallel.
Parallel vectors
Two vectors are parallel if one can be written as a scalar multiple of the other, i.e., if u=kv for some scalar k. Parallel vectors have identical or exactly opposite directions.
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Exercise
For what values of a is (2a−34) parallel to (3a+1−6)?
Select the correct option
Scaling a vector keeps its direction (or reverses its direction for a negative scalar) but changes its size. We call a vector's "size" its magnitude, and calculate it with the formula below:
Vector Magnitude
The magnitude (or length) of a vector v=⎝⎛v1v2v3⎠⎞ is calculated as
Geometrically, this represents the distance from the vector's starting point to its endpoint in 3D space. For example, the magnitude of the displacement vector AB equals the straight-line distance between points A and B.
Example
Find ∣∣∣∣∣∣⎝⎛122⎠⎞∣∣∣∣∣∣.
Discussion
The point P has coordinates (x,y).
What is the distance from P to the origin?
Consider P's position vector OP⃗, defined as
What is the magnitude of OP⃗?
Is there a connection between these two formulas?
In this context, what do you think the formula for the magnitude of a vector represents?
Part (a)
The origin has coordinates (0,0). By the distance formula, the distance d from P(x,y) to the origin is
which simplifies to
Hence the distance from P to the origin is √x2+y2.
Part (b)
The magnitude of a vector v⃗=(v1,v2) is given by √v12+v22. Hence, for OP⃗=(xy)
Part (c)
Both formulas in parts (a) and (b) are identical because they both come from the same Pythagorean distance calculation. In part (a) we found the distance from P(x,y) to the origin by
and in part (b)
In general, for any vector v⃗=(v1,v2), the formula
is just the distance between its tail (0,0) and its head (v1,v2). Hence “magnitude” really is the straight-line distance from where the vector begins to where it ends.
This connection between magnitude and distance can be generalized beyond position vectors: the distance between any two points A and B is the absolute value of the magnitude of the displacement vector AB⃗.
Checkpoint
Find the magnitude of the vector⎝⎛−42−1⎠⎞.
Select the correct option
Exercise
Two vectors v and w are defined
State the value(s) of a for which v and w have the same magnitude.
Select the correct option
Example
Given that ⎝⎛2ab⎠⎞∥⎝⎛−639⎠⎞, find a and b.
So 2=−6k⇒k=−31:
So a=−1 and b=−3.
It can be useful to represent vectors with magnitudes greater than one as multiples of a vector in the same direction with a magnitude of one. This makes it straightforward to tell what the magnitude of a non-unit vector is.
Discussion
The vector v=⎝⎛30−4⎠⎞ has a magnitude of ∣v∣=5.
Can you write the equation of a vector pointing in the same direction as v but with a magnitude of one?
We want a vector u that points in the same direction as v but has a length of 1. Any such u must be a scalar multiple of v, say
What happens to the length when we scale a vector by λ? Well, scaling multiplies every component by λ, so distances from the origin also stretch by a factor ∣λ∣. In symbols,
Since ∣∣v∣∣=5, we get
We need ∣∣u∣∣=1, so
To keep the same direction (not reverse it), pick λ=+51. Hence
As a quick check,
Unit vectors
When a vector u has a magnitude of 1, we say u is a unit vector.
We often express other vectors v with magnitude ∣v∣=1 as scalar multiples of the unit vector pointing in the same direction,
where k=∣v∣.
The unit vector in the same direction as a non-unit vector v is written as
Rescaling to form a vector with a magnitude of one is referred to as normalizing a vector.
Checkpoint
The unit vector u is defined u=⎝⎛32−3132⎠⎞. The point P has a position vector OP⃗ given by OP⃗=−4u.
Find the coordinates of the point P.
Select the correct option
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