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

Vector Operations in Cartesian Form in Two-Space

C2.1 C1.3About 70 minutes

Learning goals

I can add, subtract and multiply vectors by scalars using their Cartesian components.
I can write vectors using the standard unit vectors 𝐢 and 𝐣, and find a unit vector in any direction.
I can add vectors given by magnitude and direction by converting to components.
I can write a vector as a linear combination of two others and test vectors and points for collinearity.

Words to know

standard unit vectors
𝐢 = [1, 0] and 𝐣 = [0, 1] in two-space.

linear combination
A sum of scalar multiples of vectors, such as a𝐮 + b𝐯.

unit vector
A vector of magnitude 1; the unit vector in the direction of 𝐮 is (1/|𝐮|)𝐮.

collinear
Parallel; two vectors are collinear if one is a scalar multiple of the other. Points are collinear if they lie on one line.

Let the components do the work

In the last two lessons, every sum needed a careful sketch and the cosine and sine laws. With vectors in Cartesian form, adding, subtracting and scaling become simple arithmetic on the components. This is how computers, GPS units and game engines handle vectors, and it is the method you will use most often from now on, including in Dot and Cross Products and Lines and Planes.

The rules

For 𝐮 = [u₁, u₂], 𝐯 = [v₁, v₂] and a scalar k:

Addition: 𝐮 + 𝐯 = [u₁ + v₁, u₂ + v₂]
Subtraction: 𝐮 − 𝐯 = [u₁ − v₁, u₂ − v₂]
Scalar multiplication: k𝐮 = [ku₁, ku₂]
Magnitude: |𝐮| = √(u₁² + u₂²)

Why does adding components work? Placing 𝐯 tip to tail after 𝐮 means moving u₁ across and u₂ up, then v₁ across and v₂ up. Altogether you have moved u₁ + v₁ across and u₂ + v₂ up.

Tip to tail on a grid: the run of the resultant is the sum of the runs (5 + 2) and its rise is the sum of the rises (1 + 3).

Worked example

Example 1: Combining vectors

Given 𝐮 = [3, −2] and 𝐯 = [−1, 5], find 𝐮 + 𝐯, 𝐮 − 𝐯, 3𝐮 − 2𝐯 and |3𝐮 − 2𝐯|.

𝐮 + 𝐯 = [3 + (−1), −2 + 5] = [2, 3].
𝐮 − 𝐯 = [3 − (−1), −2 − 5] = [4, −7].
3𝐮 = [9, −6] and 2𝐯 = [−2, 10], so 3𝐮 − 2𝐯 = [9 − (−2), −6 − 10] = [11, −16].
|3𝐮 − 2𝐯| = √(121 + 256) = √377 ≈ 19.42.

[2, 3], [4, −7], [11, −16], √377 ≈ 19.42.

The standard unit vectors 𝐢 and 𝐣

Two special unit vectors point along the axes: 𝐢 = [1, 0] and 𝐣 = [0, 1]. Every 2-D vector can be built from them:

[a, b] = a𝐢 + b𝐣

For example, [3, 2] = 3𝐢 + 2𝐣 and [5, −4] = 5𝐢 − 4𝐣. Physics and engineering books often use this notation.

Unit vector in a given direction. To get a vector of length 1 pointing the same way as 𝐮, divide each component by |𝐮|. To get a vector of length L in that direction, multiply the unit vector by L.

𝐢 and 𝐣 are one unit long along the axes. The vector [3, 2] is three 𝐢 steps followed by two 𝐣 steps.

Worked example

Example 2: Unit vectors and a chosen length

Find the unit vector in the direction of 𝐮 = [3, −4], and the vector of magnitude 10 in the same direction.

|𝐮| = √(9 + 16) = 5.
Unit vector: (1/5)[3, −4] = [3/5, −4/5] = [0.6, −0.8]. Check: √(0.36 + 0.64) = 1 ✓
Length 10: 10[0.6, −0.8] = [6, −8].

[3/5, −4/5] and [6, −8].

Worked example

Example 3: Adding vectors given by magnitude and direction

𝐚 = (10, 30°) and 𝐛 = (6, 135°) in polar form. Find 𝐚 + 𝐛 in polar form.

Convert to components: 𝐚 = [10 cos 30°, 10 sin 30°] ≈ [8.660, 5.000] and 𝐛 = [6 cos 135°, 6 sin 135°] ≈ [−4.243, 4.243].
Add: 𝐚 + 𝐛 ≈ [4.418, 9.243].
Magnitude: √(4.418² + 9.243²) ≈ 10.24. Both components positive (quadrant I): θ = tan⁻¹(9.243 ÷ 4.418) ≈ 64.5°.

𝐚 + 𝐛 ≈ (10.24, 64.5°). No triangle and no sine law needed: this method works for any number of vectors.

Linear combinations and collinearity

An expression like a𝐮 + b𝐯 (scalars times vectors, added) is a linear combination of 𝐮 and 𝐯. If 𝐮 and 𝐯 are not collinear, every vector in the plane can be written as a linear combination of them in exactly one way. To find the scalars, set up one equation per component and solve the system.

Two vectors are collinear when one is a scalar multiple of the other. In components, [a, b] and [c, d] are collinear when the ratios of matching components agree (for non-zero components), for example [6, −9] = −3[−2, 3].

Worked example

Example 4: Writing a vector as a linear combination

Write [8, 1] as a linear combination of [1, 2] and [2, −1].

We want a[1, 2] + b[2, −1] = [8, 1], so a + 2b = 8 and 2a − b = 1.
From the second equation, b = 2a − 1. Substitute: a + 2(2a − 1) = 8, so 5a = 10 and a = 2. Then b = 3.
Check: 2[1, 2] + 3[2, −1] = [2 + 6, 4 − 3] = [8, 1] ✓

[8, 1] = 2[1, 2] + 3[2, −1]

The linear combination 2[1, 2] + 3[2, −1], drawn tip to tail, lands exactly on [8, 1].

Worked example

Example 5: Testing for collinear points

Are A(−1, 4), B(1, 1) and C(5, −5) collinear?

𝐀𝐁 = [2, −3] and 𝐀𝐂 = [6, −9].
𝐀𝐂 = 3[2, −3] = 3𝐀𝐁, so the two vectors are parallel. They also share the point A, so they lie on the same line.

Yes, A, B and C are collinear (and B is one third of the way from A to C).

Just for fun

Many games move a character by adding a velocity vector to its position every frame, often about 60 times a second. So while you play, your computer is doing this lesson's homework dozens of times a second, without complaining once.

Hands-on activity

Animate a jump on graph paper

You need: graph paper (or a printed grid), a pencil and a ruler.

Game engines move objects with vector addition. Start a ball at position P₀ = (0, 0) with velocity 𝐯₀ = [2, 3]. Each "frame":

New position = old position + velocity.
New velocity = old velocity + [0, −1] (a simple "gravity" vector).

Repeat for 7 frames, plotting each position, then connect the dots.

What to look for: the positions are (2, 3), (4, 5), (6, 6), (8, 6), (10, 5), (12, 3) and (14, 0), and the velocities go [2, 2], [2, 1], [2, 0], [2, −1], … The dots trace a parabola-shaped path, symmetric about x = 7, like a real jump. Try changing the gravity vector to [0, −2] and see how the path changes.

Your turn

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

Find 𝐚 − 𝐛.

Find 4𝐚.

Find 2𝐚 + 3𝐛.

Find |𝐚 − 𝐛|.

Find the unit vector in the direction of [−5, 12].

Write [7, −3] in terms of 𝐢 and 𝐣.

More practice: warm-up

[1, 2] + [3, 4] = ?

[5, 1] − [2, 7] = ?

−2[3, −1] = ?

Write 3𝐢 + 𝐣 in Cartesian form.

Write the zero vector in Cartesian form.

More practice: on your own

Find x so that [x, 6] and [2, −3] are collinear.

Solve for the vector 𝐱: 2𝐱 + [1, −4] = [7, 2].

Write [1, 12] as a linear combination of [2, 3] and [−1, 2].

Find the vector of magnitude 15 that points in the direction opposite to [−3, 4].

Three forces act on a point: 50 N at 0°, 80 N at 120° and 40 N at 225° (standard angles). Find the resultant.

Challenge

Show that every vector [a, b] can be written as a linear combination of [1, 1] and [1, −1], and find the scalars.

Find every k for which |[k, k + 1]| = 5.

A is (2, 1) and B is (6, 4). Find the point P on segment AB with AP : PB = 2 : 3.

Check your understanding

Quiz

What is [4, −1] + [−6, 3]?

What is 3[2, −5] − [1, 4]?

Which is the unit vector in the direction of [6, −8]?

How is [−2, 7] written with 𝐢 and 𝐣?

Which vector is collinear with [4, −6]?

What is |2𝐢 − 𝐣|?

For parents and teachers

Big idea (C2.1, C1.3): in Cartesian form, vector operations are done component by component. Converting magnitude-direction vectors to components first makes even messy sums simple.

How to help: The graph-paper "jump" activity is a good one to do together; it shows why computer games run on vector addition. When checking homework, ask your teen to estimate the answer from a quick sketch first.

Common difficulties:

Sign errors when subtracting negative components: 3 − (−1) = 4.
Forgetting to multiply both components by the scalar.
Mixing up i/j with x/y order when writing a𝐢 + b𝐣.
Rounding converted components too early; keep at least three decimal places until the end.