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

Vectors in Two-Space and Three-Space: Unit Review

C1.1 C1.2 C1.3 C1.4 C2.1 C2.2 C2.3About 40 minutes

Learning goals

I can represent vectors geometrically and algebraically in two-space and three-space, and convert between forms.
I can add, subtract and multiply vectors by scalars and use the properties of these operations.
I can solve force, velocity and geometry problems with vectors.

Words to know

vector
A quantity with magnitude and direction.

resultant
The sum of two or more vectors.

unit vector
A vector of magnitude 1, such as (1/|𝐮|)𝐮.

Let's review!

In this unit you learned that a vector has magnitude and direction, and you met vectors in many forms: arrows (directed line segments), bearings and standard angles, Cartesian components [x, y] and [x, y, z], and polar form (r, θ). You converted between forms with trigonometry, moved into three-space, and learned to add, subtract and scale vectors both geometrically and with components. You proved the properties of these operations and used them to solve problems about forces, velocities and geometry.

The next unit, Dot and Cross Products, introduces two ways of multiplying vectors together, used to find angles, projections, work, areas and torque. After that, Lines and Planes uses vectors to describe lines and planes in three-space. This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Work through one or two sections at a time, with a calculator in degree mode. Draw a sketch for every question: a compass cross for directions, a tip-to-tail triangle for sums, a box for 3-D points. Keep full calculator values until the end, then round (magnitudes to one decimal place and angles to the nearest tenth of a degree unless the question says otherwise). For each question you miss, write down why (wrong quadrant? wrong angle in the cosine law? wind direction backwards?) and go back to that lesson. Finish with the 10-question quiz.

How the unit fits together. Representing vectors (top) leads to the operations (middle), whose properties and applications (bottom) finish the unit. Numbers in brackets are lesson numbers.

Two formulas worth remembering. With θ the angle between 𝐮 and 𝐯 placed tail to tail, the cosine law gives the lengths of both diagonals of the parallelogram.

Just for fun

You now know that "Which way to the arena?" deserves a vector for an answer. Use this power responsibly.

Vectors: Quantities with Magnitude and Direction

Classify as scalar or vector: (a) 70 km/h [S]; (b) 3 h; (c) a 12 N force straight up.

Ali walks 9 m west and then 12 m north. Find the distance walked and the displacement.

Why doesn't |𝐮| = |𝐯| mean 𝐮 = 𝐯?

Drawing Vectors: Directed Line Segments, Bearings and Equal Vectors

Write the true bearing 145° as a quadrant bearing and as a standard angle.

Write N 15° W as a true bearing and a standard angle.

ABCD is a parallelogram. Which vector, named with its letters, equals 𝐁𝐂?

Vectors in Cartesian and Polar Form

Find 𝐀𝐁 and |𝐀𝐁| for A(4, −1) and B(−2, 7).

A is (3, 5) and 𝐀𝐁 = [−4, 2]. Find B.

Write [0, −5] in polar form.

Converting Between Cartesian Form and Magnitude-Direction Form

Write (12, 135°) in Cartesian form, exactly.

Write [−7, −24] in magnitude-direction form (standard angle).

A velocity is 30 km/h on a bearing of 240°. Write it in Cartesian form (x east, y north).

Adding Vectors Geometrically: Triangle and Parallelogram Laws

Simplify 𝐀𝐁 + 𝐁𝐂 + 𝐂𝐃 + 𝐃𝐄.

|𝐮| = 7 and |𝐯| = 24, and they are perpendicular. Find |𝐮 + 𝐯|.

|𝐮| = 8, |𝐯| = 6 and the angle between them is 120°. Find |𝐮 + 𝐯|.

Subtracting Vectors and Multiplying by a Scalar

For the same 𝐮 and 𝐯 (|𝐮| = 8, |𝐯| = 6, angle 120°), find |𝐮 − 𝐯|.

𝐮 = 4 N [N 10° E]. Describe −3𝐮.

In parallelogram ABCD with 𝐚 = 𝐀𝐁 and 𝐝 = 𝐀𝐃, write 𝐁𝐃.

Vector Operations in Cartesian Form in Two-Space

Find 2[3, −1] − 3[−1, 4].

Find the unit vector in the direction of [8, −15].

Write [5, 0] as a linear combination of [1, 1] and [1, −1].

Points and Vectors in Three-Space

Find the distance between (1, −1, 2) and (3, 5, −1).

Find |[6, −2, 3]|.

In which coordinate plane is (0, 4, −2)?

Vector Operations in Three-Space

Find [1, 0, −2] + 2[3, −1, 1].

Are A(0, 1, 2), B(2, 2, 5) and C(6, 4, 11) collinear?

Find the unit vector in the direction of 𝐢 − 2𝐣 + 2𝐤.

Properties of Vector Addition and Scalar Multiplication

Simplify 4(𝐮 − 2𝐯) − 3(2𝐮 − 𝐯).

Solve 5𝐱 − 2𝐮 = 3𝐱 + 4𝐯 for 𝐱.

Name the property: (3 + 4)𝐮 = 3𝐮 + 4𝐮.

Applications: Forces, Resultants and Equilibrants

Find the resultant and the equilibrant of 20 N [E] and 15 N [N].

A 40 kg box sits on a 15° ramp. Find the components of its weight along and perpendicular to the ramp.

A 10 kg sign hangs from two cables, each making 30° with the horizontal. Find the tension in each cable.

Applications: Velocity, Wind and Current

A plane heads east at 300 km/h with a wind from the north at 50 km/h. Find the ground velocity.

A boat heads straight across a 150 m river at 5 m/s; the current is 2 m/s. Find the crossing time and the drift.

For the same boat, in what direction should it head to land directly opposite?

Solving Geometry Problems with Vectors

Find the midpoint of (−2, 4, 6) and (4, 0, −2).

Find P on segment AB with AP : PB = 1 : 3, for A(0, 0, 0) and B(8, 4, −12).

In △ABC, M and N are midpoints of AB and AC. What does 𝐌𝐍 = ½𝐁𝐂 tell you?

Check your understanding

Quiz

Which quantity is a vector?

What is the true bearing of S 40° W?

What is the vector from A(1, 4) to B(−2, 8)?

What is the Cartesian form of (10, 210°)?

|𝐮| = 5 and |𝐯| = 12, and 𝐮 ⊥ 𝐯. What is |𝐮 + 𝐯|?

What is 2[1, −3, 2] − [4, 0, −1]?

What is the distance from the origin to (2, 3, 6)?

Which property is shown by 𝐮 + 𝐯 = 𝐯 + 𝐮?

What is the equilibrant of the force [5, −12] N?

10.

A plane heads north at an airspeed of 400 km/h with a wind from the west at 30 km/h. What is its ground speed?

For parents and teachers

Use the review to find what to revisit:

Mixing up scalars and vectors, or distance and displacement → Lesson 1 (Vectors: Quantities with Magnitude and Direction).
Bearings measured the wrong way, or confusion about equal vectors → Lesson 2 (Drawing Vectors: Directed Line Segments, Bearings and Equal Vectors).
Subtracting tail minus head, or polar form with a negative r → Lesson 3 (Vectors in Cartesian and Polar Form).
Wrong quadrant from tan⁻¹, or sin/cos swapped for bearings → Lesson 4 (Converting Between Cartesian Form and Magnitude-Direction Form).
Wrong angle in the cosine law for a sum → Lesson 5 (Adding Vectors Geometrically).
𝐮 − 𝐯 drawn backwards, or negative magnitudes → Lesson 6 (Subtracting Vectors and Multiplying by a Scalar).
Arithmetic slips with components, unit vectors or linear combinations → Lesson 7 (Vector Operations in Cartesian Form in Two-Space).
Trouble plotting 3-D points or using the 3-D distance formula → Lesson 8 (Points and Vectors in Three-Space).
Collinearity or linear-combination errors in three-space → Lesson 9 (Vector Operations in Three-Space).
Sign errors when simplifying, or "dividing by a vector" → Lesson 10 (Properties of Vector Addition and Scalar Multiplication).
Equilibrant in the wrong direction, or tension problems → Lesson 11 (Applications: Forces, Resultants and Equilibrants).
Wind direction backwards, or heading confused with track → Lesson 12 (Applications: Velocity, Wind and Current).
Ratio m : n used as m/n, or proofs that rely on a picture → Lesson 13 (Solving Geometry Problems with Vectors).

Looking ahead: the next unit, Dot and Cross Products, uses these skills to find angles between vectors, projections, work, areas, volumes and torque.