C. Geometry and Algebra of Vectors · Vectors in Two-Space and Three-Space

Vectors in Cartesian and Polar Form

C1.2 C1.3About 65 minutes

Learning goals

I can write a vector in two-space in Cartesian form [a, b] and as a position vector.
I can find the vector between two points (head minus tail) and its magnitude.
I can describe a vector in polar form (r, θ) and recognize equal vectors from their components.
I can use equal vectors to find unknown points and components.

Words to know

Cartesian form
A vector written by its components, such as [4, 3]: 4 units in the x-direction and 3 in the y-direction.

components
The x- and y-parts of a vector in Cartesian form.

position vector
The vector 𝐎𝐏 from the origin to a point P; for P(a, b) it is [a, b].

polar form
A vector written as (r, θ), where r is its magnitude and θ its standard angle.

magnitude formula
|[a, b]| = √(a² + b²).

Arrows you can calculate with

Drawings are great for seeing vectors, but they are slow and not very precise. If we put vectors on a coordinate grid, we can describe each one with just two numbers and let algebra do the work. This lesson introduces the two algebraic forms of a 2-D vector that the curriculum asks for: Cartesian form (components) and polar form (magnitude and angle).

Cartesian form

Put the tail of a vector at the origin O. Its head lands at some point P(a, b). The vector 𝐎𝐏 is called the position vector of P, and we write it in Cartesian form as

𝐎𝐏 = [a, b]

The numbers a and b are the x-component and y-component: how far the vector goes right (or left, if negative) and up (or down).

Many Ontario textbooks use square brackets [a, b] for a vector and round brackets (a, b) for a point. Some books, and the curriculum itself, write vectors with round brackets, as in "the vector (8, 6)". Either is fine if you are consistent; on this site, vectors get square brackets.

Because a vector can slide anywhere, every vector equals some position vector. To find its components, count the run and the rise from tail to head.

The position vector 𝐎𝐏 and the vector 𝐀𝐁 have the same components [4, 3], so they are equal vectors, even though they sit in different places.

The vector between two points

If A(x₁, y₁) is the tail and B(x₂, y₂) is the head, then

𝐀𝐁 = [x₂ − x₁, y₂ − y₁] (head minus tail)

and its magnitude, by the Pythagorean theorem, is

|𝐀𝐁| = √((x₂ − x₁)² + (y₂ − y₁)²)

You will recognize this as the distance formula from Grade 10. In general, |[a, b]| = √(a² + b²).

Two vectors in Cartesian form are equal exactly when their components are equal: [a, b] = [c, d] means a = c and b = d.

Worked example

Example 1: Components and magnitude

Given A(−2, 1) and B(4, 9), find 𝐀𝐁, |𝐀𝐁| and 𝐁𝐀.

Head minus tail: 𝐀𝐁 = [4 − (−2), 9 − 1] = [6, 8].
|𝐀𝐁| = √(6² + 8²) = √100 = 10.
𝐁𝐀 goes the other way: [−2 − 4, 1 − 9] = [−6, −8] = −𝐀𝐁.

𝐀𝐁 = [6, 8], |𝐀𝐁| = 10, 𝐁𝐀 = [−6, −8].

Worked example

Example 2: Using equal vectors to find a point

With A(−2, 1) and B(4, 9) as above, and C(1, −3), find the point D so that 𝐂𝐃 = 𝐀𝐁.

Let D = (x, y). Then 𝐂𝐃 = [x − 1, y − (−3)] = [x − 1, y + 3].
Set the components equal to [6, 8]: x − 1 = 6 and y + 3 = 8.
So x = 7 and y = 5. Shortcut: D = C + (the components of 𝐀𝐁) = (1 + 6, −3 + 8).

D(7, 5)

Polar form

A vector can also be given by its magnitude r and its standard angle θ (anticlockwise from the positive x-axis). This is polar form, written (r, θ), for example (10, 36.9°). The degree sign (or the word "rad" for radians, which you met in MHF4U) makes it clear that the second number is an angle.

(5, 0°) points along the positive x-axis: it is [5, 0].
(4, 90°) points straight up: [0, 4].
(5, 180°) points left: [−5, 0].
(6, 270°) points straight down: [0, −6].

Directions given as bearings (such as 320° or N 40° W) describe the same idea; convert them to a standard angle first. In the next lesson you will convert between polar and Cartesian form for any angle using trigonometry.

The vector [−3, 3] has magnitude r = √(3² + 3²) = 3√2 and makes a 45° angle with the negative x-axis, so θ = 180° − 45° = 135°.

Worked example

Example 3: Polar form with a special triangle

Write [−3, 3] in polar form, with an exact magnitude.

r = √((−3)² + 3²) = √18 = 3√2.
The arrow is in quadrant II. Its run and rise have equal size, so it makes a 45° angle with the negative x-axis (a 45-45-90 triangle).
θ = 180° − 45° = 135°.

(3√2, 135°) (about (4.24, 135°)).

Worked example

Example 4: An unknown component

Find every value of k for which |[k, 12]| = 13.

√(k² + 12²) = 13, so k² + 144 = 169.
k² = 25, so k = 5 or k = −5.
Check: √(25 + 144) = √169 = 13 ✓ (for both).

k = 5 or k = −5

Head minus tail

The most common slip is subtracting in the wrong order. Say it out loud: "head minus tail". If you get the order backwards you get −𝐀𝐁 instead of 𝐀𝐁: the right size but the opposite direction. A quick sketch shows whether the components should be positive or negative.

Just for fun

A vector walks into a coordinate plane and says, "Take me to your origin." The plane replies, "Which one? You're free to start anywhere."

Hands-on activity

Tile-floor vectors

You need: a floor with square tiles (or a chessboard, or a sheet of graph paper), a coin and a button (or any two small markers), and paper.

Choose one tile corner as the origin O and agree that "right" is +x and "away from you" is +y.
Place the coin at a tile corner. Write its position vector 𝐎𝐂 in Cartesian form by counting tiles (for example [3, −2]).
Place the button somewhere else and write 𝐎𝐁.
Find the vector from the coin to the button by counting, then check with "head minus tail".
Measure one tile with a ruler, and use it to work out the real distance between the coin and the button with the magnitude formula. Then measure it directly with the ruler.

What to look for: the counted vector should equal 𝐎𝐁 − 𝐎𝐂 component by component. The measured and calculated distances should agree to within a few millimetres per tile.

Your turn

Write the position vector of P(−5, 2) and find its magnitude.

Given A(3, −1) and B(−2, 5), find 𝐀𝐁 and |𝐀𝐁|.

For the same points, write 𝐁𝐀.

A is (1, 2) and 𝐀𝐁 = [3, −7]. Find B.

Is 𝐀𝐁 = 𝐂𝐃 for A(0, 0), B(2, 5), C(−3, 1), D(−1, 6)?

Write (6, 270°) in Cartesian form.

Find every value of k for which |[k, −8]| = 10.

More practice: warm-up

Find |[3, 4]|.

Write the position vector of (7, −2).

Find the vector from (1, 1) to (4, 5).

Write (2, 0°) in Cartesian form.

Find |[0, −9]|.

More practice: on your own

Write [−2√3, 2] in polar form.

A(1, 1), B(5, 2) and C(6, 6) are three vertices of parallelogram ABCD. Find D.

Show that A(−1, 2), B(3, 5) and C(6, 1) form an isosceles right triangle.

Write four different vectors with integer components and magnitude 5.

M is the midpoint of A(2, −3) and B(8, 5). Find M and show that 𝐀𝐌 = 𝐌𝐁.

Challenge

A is (3, 6). Find every point P on the x-axis with |𝐀𝐏| = 10.

Find a and b if [a + 2, 3] = [5, b − 1].

Prove, using coordinates, that if 𝐀𝐁 = 𝐂𝐃 then 𝐀𝐂 = 𝐁𝐃.

Check your understanding

Quiz

What is the vector from A(2, 5) to B(−1, 9)?

What is |[−6, 8]|?

Which Cartesian vector equals the polar vector (3, 180°)?

A is (1, −2) and 𝐀𝐁 = [5, 3]. What is B?

Which vector equals [2, −3]?

Which is the polar form of [0, −4]?

For parents and teachers

Big idea (C1.2, C1.3): a vector on a grid can be written as two numbers, its components [a, b], or as a length and an angle, (r, θ). The vector between two points is "head minus tail".

How to help: Give your teen two points (for example two houses on a street map with a grid) and ask for the vector from one to the other, then the reverse vector. They should notice that the components just change sign.

Common difficulties:

Subtracting tail minus head (gives the opposite vector).
Confusing a point (a, b) with a vector [a, b]. A point is a location; a vector is a displacement.
Writing a negative magnitude in polar form, such as (−4, 90°), instead of (4, 270°).
Sign errors when squaring negatives: (−6)² = 36, not −36.