Vectors and matrices

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1. Vectors and their components

Discover

You walk from P=(1,2)P = (1, 2) to Q=(4,6)Q = (4, 6). How far do you go to the right, and how far up?

  1. 3 to the right and 4 up
  2. 3 to the left and 4 down

Key idea

A vector has a length and a direction, but no fixed place. We write it by its components.

From P=(x1,y1)P = (x_1, y_1) to Q=(x2,y2)Q = (x_2, y_2): PQ→=⟨x2−x1, y2−y1⟩\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle (end minus start). Two vectors with the same components are equal, wherever they start.

Worked example

Find PQ→\overrightarrow{PQ} for P=(5,−1)P = (5, -1) and Q=(2,3)Q = (2, 3).

2−5=−32 - 5 = -3
End minus start, horizontally.
3−(−1)=43 - (-1) = 4
End minus start, vertically.
PQ→=⟨−3,4⟩\overrightarrow{PQ} = \langle -3, 4 \rangle
Write the components.

Your turn, guided

Your turn: P=(2,7)P = (2, 7) and Q=(6,1)Q = (6, 1).

Horizontal: 44
6 − 2.
Vertical: −6-6, so PQ→=⟨4,−6⟩\overrightarrow{PQ} = \langle 4, -6 \rangle
1 − 7.

Practice ★

  1. P=(2,4)P = (2, 4) and Q=(4,−6)Q = (4, -6). Find the components of PQ→\overrightarrow{PQ}.
  2. P=(5,1)P = (5, 1) and Q=(7,−2)Q = (7, -2). Find the components of PQ→\overrightarrow{PQ}.
  3. P=(2,3)P = (2, 3) and Q=(9,−8)Q = (9, -8). Find the components of PQ→\overrightarrow{PQ}.

Watch out!

Practice ★★

  1. The vector ⟨2,1⟩\langle 2, 1 \rangle starts at the point (1,1)(1, 1). Tap the point where it ends.
  2. The vector ⟨2,3⟩\langle 2, 3 \rangle starts at the point (1,−3)(1, -3). Tap the point where it ends.
  3. The vector ⟨3,1⟩\langle 3, 1 \rangle starts at the point (−2,−1)(-2, -1). Tap the point where it ends.
  1. P=(3,3)P = (3, 3) and Q=(8,8)Q = (8, 8). Which vector is PQ→\overrightarrow{PQ}?
    1. ⟨5,5⟩\langle 5, 5 \rangle
    2. ⟨−5,−5⟩\langle -5, -5 \rangle
    3. ⟨11,11⟩\langle 11, 11 \rangle
  2. P=(2,2)P = (2, 2) and Q=(8,6)Q = (8, 6). Which vector is PQ→\overrightarrow{PQ}?
    1. ⟨6,4⟩\langle 6, 4 \rangle
    2. ⟨−6,−4⟩\langle -6, -4 \rangle
    3. ⟨10,8⟩\langle 10, 8 \rangle

Exit ticket

Three questions to check you have got it.

  1. P=(2,4)P = (2, 4) and Q=(4,−6)Q = (4, -6). Find the components of PQ→\overrightarrow{PQ}.
  2. P=(5,1)P = (5, 1) and Q=(7,−2)Q = (7, -2). Find the components of PQ→\overrightarrow{PQ}.
  3. P=(2,3)P = (2, 3) and Q=(9,−8)Q = (9, -8). Find the components of PQ→\overrightarrow{PQ}.

Summary card

Vectors and components

  • PQ→=⟨x2−x1, y2−y1⟩\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle: end minus start.
  • Same components, same vector, wherever it starts.

Watch out: start minus end gives the opposite vector.

2. Vectors in context: velocity and force

Warm-up

Three quick questions from earlier lessons.

Discover

A swimmer swims north at 3 km/h across a river flowing east at 4 km/h. How fast does she move over the ground?

  1. 5 km/h
  2. 7 km/h

Key idea

Velocities and forces are vectors: when several act together, add them.

The resultant of several vectors is their sum. Its magnitude is the speed (or total force), and its direction angle is where the object actually goes. Write each vector in components first.

Worked example

Forces of 30 N due east and 40 N due north pull on a ring. Find the resultant.

F⃗=⟨30,40⟩\vec{F} = \langle 30, 40 \rangle
Components.
∣F⃗∣=50|\vec{F}| = 50 N
√(900 + 1600).
θ=tan⁡−143≈53.1∘\theta = \tan^{-1}\dfrac{4}{3} \approx 53.1^\circ north of east
tan θ = 40/30.

Your turn, guided

Your turn: a plane flies east at 200 mph with a wind from the south at 50 mph.

Ground velocity ⟨200,50⟩\langle 200, 50 \rangle
East and north components.
Speed =42500≈206.2= \sqrt{42500} \approx 206.2 mph
√(40000 + 2500) ≈ 206.2: type the exact value.

Practice ★

  1. A plane flies due east at 101 miles per hour through the air, and the wind blows due north at 32 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  2. A plane flies due east at 236 miles per hour through the air, and the wind blows due north at 60 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  3. A plane flies due east at 174 miles per hour through the air, and the wind blows due north at 17 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale

Watch out!

Practice ★★

  1. Two people pull a box: one with 36 newtons due east, the other with 59 newtons due north. At what angle north of east does the box start to move? Give degrees to 1 decimal place.Not to scale
  2. Two people pull a box: one with 27 newtons due east, the other with 23 newtons due north. At what angle north of east does the box start to move? Give degrees to 1 decimal place.Not to scale
  3. Two people pull a box: one with 88 newtons due east, the other with 72 newtons due north. At what angle north of east does the box start to move? Give degrees to 1 decimal place.Not to scale

Exit ticket

Three questions to check you have got it.

  1. A plane flies due east at 101 miles per hour through the air, and the wind blows due north at 32 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  2. A plane flies due east at 236 miles per hour through the air, and the wind blows due north at 60 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale
  3. A plane flies due east at 174 miles per hour through the air, and the wind blows due north at 17 miles per hour. Find its speed over the ground, to 1 decimal place.Not to scale

Summary card

Vectors in context

  1. Write each velocity or force in components.
  2. Add them: the resultant.
  3. Magnitude = speed or total force; tan⁡θ\tan\theta = direction.

Watch out: speeds in different directions do not simply add.

3. Matrices: representing data, adding and multiplying by a scalar

Warm-up

Three quick questions from earlier lessons.

Discover

A table has 2 rows and 3 columns of numbers. How many numbers does it hold?

  1. 6
  2. 5

Key idea

A matrix is a rectangular table of numbers.

An m×nm \times n matrix has mm rows and nn columns; the entry in row ii, column jj is aija_{ij}. Matrices of the same size add and subtract entry by entry, and a scalar kk multiplies every entry: 3(120−1)=(360−3)3\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 0 & -3 \end{pmatrix}.

Worked example

Add (1234)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} and (50−12)\begin{pmatrix} 5 & 0 \\ -1 & 2 \end{pmatrix}.

1 + 5, 2 + 0, 3 + (−1), 4 + 2
Same positions.
(6226)\begin{pmatrix} 6 & 2 \\ 2 & 6 \end{pmatrix}
Result.

Your turn, guided

Your turn: the top-left entry of 4(2103)\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}.

88
4 · 2.

Practice ★

  1. Let A = (3−375)\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix} and B = (−4−4−55)\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}. What is the entry in row 1, column 2 of A + B?
  2. Let A = (2761)\begin{pmatrix} 2 & 7 \\ 6 & 1 \end{pmatrix} and B = (5−555)\begin{pmatrix} 5 & -5 \\ 5 & 5 \end{pmatrix}. What is the entry in row 1, column 2 of A + B?
  1. A shop records sales in a matrix with one row for each of its 4 stores and one column for each of 7 days. What are the dimensions of the matrix?
    1. 4 × 7
    2. 7 × 4
    3. 28 × 1

Watch out!

Practice ★★

  1. Let A = (2062)\begin{pmatrix} 2 & 0 \\ 6 & 2 \end{pmatrix}. What is the entry in row 2, column 1 of 3A?
  2. Let A = (82−1−3)\begin{pmatrix} 8 & 2 \\ -1 & -3 \end{pmatrix}. What is the entry in row 2, column 1 of 3A?
  1. Let A = (3−375)\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix} and B = (−4−4−55)\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}. What is the entry in row 1, column 2 of A + B?

Exit ticket

Three questions to check you have got it.

  1. Let A = (3−375)\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix} and B = (−4−4−55)\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}. What is the entry in row 1, column 2 of A + B?
  2. Let A = (2761)\begin{pmatrix} 2 & 7 \\ 6 & 1 \end{pmatrix} and B = (5−555)\begin{pmatrix} 5 & -5 \\ 5 & 5 \end{pmatrix}. What is the entry in row 1, column 2 of A + B?
  3. Let A = (4618)\begin{pmatrix} 4 & 6 \\ 1 & 8 \end{pmatrix} and B = (7259)\begin{pmatrix} 7 & 2 \\ 5 & 9 \end{pmatrix}. What is the entry in row 1, column 2 of A + B?

Summary card

Matrices

Size is rows × columns; add entry by entry; a scalar multiplies every entry.

Watch out: rows first.

4. Multiplying matrices and its properties

Warm-up

Three quick questions from earlier lessons.

Discover

To multiply a row (2, 3) by a column (4, 5), you compute 2 · 4 + 3 · 5. What is it?

  1. 23
  2. (8, 15)

Key idea

Each entry of AB is a row of A times a column of B.

If AA is m×nm \times n and BB is n×pn \times p, then ABAB is m×pm \times p: the entry in row ii, column jj is row ii of AA times column jj of BB (multiply pairs, then add). The inner sizes must match. Matrix multiplication is associative and distributes over addition, but in general AB≠BAAB \ne BA.

Worked example

Compute the top-left entry of (1234)(5678)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}.

(1, 2) and (5, 7)
Row 1 of A, column 1 of B.
1 · 5 + 2 · 7 = 19
Multiply and add.

Your turn, guided

Your turn: the top-right entry of the same product.

2222
1 · 6 + 2 · 8.

Practice ★

  1. Let A = (423−2)\begin{pmatrix} 4 & 2 \\ 3 & -2 \end{pmatrix} and B = (7954)\begin{pmatrix} 7 & 9 \\ 5 & 4 \end{pmatrix}. What is the entry in row 1, column 1 of AB?
  2. Let A = (−2−45−4)\begin{pmatrix} -2 & -4 \\ 5 & -4 \end{pmatrix} and B = (47−5−1)\begin{pmatrix} 4 & 7 \\ -5 & -1 \end{pmatrix}. What is the entry in row 1, column 1 of AB?
  1. A is a 2 × 5 matrix and B is a 5 × 8 matrix. What are the dimensions of AB?
    1. 2 × 8
    2. 5 × 5
    3. AB is not defined

Watch out!

Practice ★★

  1. Let A = (8884)\begin{pmatrix} 8 & 8 \\ 8 & 4 \end{pmatrix} and B = (2332)\begin{pmatrix} 2 & 3 \\ 3 & 2 \end{pmatrix}. What is the entry in row 2, column 1 of AB?
  2. Let A = (182−2)\begin{pmatrix} 1 & 8 \\ 2 & -2 \end{pmatrix} and B = (2−556)\begin{pmatrix} 2 & -5 \\ 5 & 6 \end{pmatrix}. What is the entry in row 2, column 1 of AB?
  1. Boss problem. A stand sells pens and notebooks. The sales matrix S = (9537)\begin{pmatrix} 9 & 5 \\ 3 & 7 \end{pmatrix} has rows Monday, Tuesday and columns pens, notebooks. The price vector is p = (36)\begin{pmatrix} 3 \\ 6 \end{pmatrix} (dollars). The product Sp gives each day's income.

Exit ticket

Three questions to check you have got it.

  1. Let A = (423−2)\begin{pmatrix} 4 & 2 \\ 3 & -2 \end{pmatrix} and B = (7954)\begin{pmatrix} 7 & 9 \\ 5 & 4 \end{pmatrix}. What is the entry in row 1, column 1 of AB?
  2. Let A = (−2−45−4)\begin{pmatrix} -2 & -4 \\ 5 & -4 \end{pmatrix} and B = (47−5−1)\begin{pmatrix} 4 & 7 \\ -5 & -1 \end{pmatrix}. What is the entry in row 1, column 1 of AB?
  3. Let A = (5804)\begin{pmatrix} 5 & 8 \\ 0 & 4 \end{pmatrix} and B = (41−23)\begin{pmatrix} 4 & 1 \\ -2 & 3 \end{pmatrix}. What is the entry in row 1, column 1 of AB?

Summary card

Multiplying matrices

(m × n)(n × p) = m × p; entry = row · column.

Watch out: AB is usually not BA.

5. Matrices as transformations: matrix times vector, determinant and area

Warm-up

Three quick questions from earlier lessons.

Discover

Doubling both coordinates of every point of a square with area 1 gives a square with area…

  1. 4
  2. 2

Key idea

A matrix moves vectors; its determinant scales areas.

Multiplying a column vector by a matrix gives a new vector: (abcd)(xy)=(ax+bycx+dy)\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}. So a 2×22 \times 2 matrix is a transformation of the plane; its columns are the images of (1,0)(1, 0) and (0,1)(0, 1). Areas are multiplied by ∣det⁡A∣|\det A|.

Worked example

Apply (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} to (3, 0) and (0, 3).

(0 · 3 − 1 · 0, 1 · 3 + 0) = (0, 3)
First point.
(−3, 0)
Second point.
a 90° counterclockwise rotation; det = 1, areas unchanged
Conclusion.

Your turn, guided

Your turn: the area scale factor of (3002)\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}.

66
det = 3 · 2.

Practice ★

  1. Let A = (9236)\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}. What is the first component of A(−3−1)\begin{pmatrix} -3 \\ -1 \end{pmatrix}?
  2. Let A = (−15−5−5)\begin{pmatrix} -1 & 5 \\ -5 & -5 \end{pmatrix}. What is the first component of A(3−2)\begin{pmatrix} 3 \\ -2 \end{pmatrix}?
  1. The matrix (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} maps the point (8, 0) to (0, 8) and (0, 8) to (−8, 0). Which transformation does it represent?
    1. A rotation of 90° counterclockwise about the origin
    2. A reflection in the line y = x
    3. A rotation of 90° clockwise about the origin

Watch out!

Practice ★★

  1. A shape with an area of 2 square units is transformed by the matrix (2223)\begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}. What is the area of the image?
  2. A shape with an area of 8 square units is transformed by the matrix (2213)\begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}. What is the area of the image?
  1. Let A = (9236)\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}. What is the first component of A(−3−1)\begin{pmatrix} -3 \\ -1 \end{pmatrix}?

Exit ticket

Three questions to check you have got it.

  1. Let A = (9236)\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}. What is the first component of A(−3−1)\begin{pmatrix} -3 \\ -1 \end{pmatrix}?
  2. Let A = (−15−5−5)\begin{pmatrix} -1 & 5 \\ -5 & -5 \end{pmatrix}. What is the first component of A(3−2)\begin{pmatrix} 3 \\ -2 \end{pmatrix}?
  3. Let A = (7−1−27)\begin{pmatrix} 7 & -1 \\ -2 & 7 \end{pmatrix}. What is the first component of A(61)\begin{pmatrix} 6 \\ 1 \end{pmatrix}?

Summary card

Matrices as transformations

Av moves vectors; columns are the images of (1, 0) and (0, 1); areas scale by |det|.

Watch out: the area factor is the determinant, not a sum of entries.

Answers: Vectors and matrices

1. Vectors and their components

A

  1. −10-10
  2. −3-3
  3. −11-11

B

  1. (3,2)(3, 2)
  2. (3,0)(3, 0)
  3. (1,0)(1, 0)

C

  1. ⟨5,5⟩\langle 5, 5 \rangle
  2. ⟨6,4⟩\langle 6, 4 \rangle

D

  1. −10-10
  2. −3-3
  3. −11-11

2. Vectors in context: velocity and force

A

  1. 54495 \sqrt{449}
  2. 437064 \sqrt{3706}
  3. 30565\sqrt{30565}

B

  1. 180tan⁡−1(5936)π\frac{180 \tan^{-1}{\left(\frac{59}{36} \right)}}{\pi}
  2. 180tan⁡−1(2327)π\frac{180 \tan^{-1}{\left(\frac{23}{27} \right)}}{\pi}
  3. 180tan⁡−1(911)π\frac{180 \tan^{-1}{\left(\frac{9}{11} \right)}}{\pi}

C

  1. 54495 \sqrt{449}
  2. 437064 \sqrt{3706}
  3. 30565\sqrt{30565}

3. Matrices: representing data, adding and multiplying by a scalar

A

  1. −7-7
  2. 22

B

  1. 4 × 7

C

  1. 1818
  2. −3-3

D

  1. −7-7

E

  1. −7-7
  2. 22
  3. 88

4. Multiplying matrices and its properties

A

  1. 3838
  2. 1212

B

  1. 2 × 8

C

  1. 2828
  2. −6-6

D

  1. 114114

E

  1. 3838
  2. 1212
  3. 44

5. Matrices as transformations: matrix times vector, determinant and area

A

  1. −29-29
  2. −13-13

B

  1. A rotation of 90° counterclockwise about the origin

C

  1. 44
  2. 3232

D

  1. −29-29

E

  1. −29-29
  2. −13-13
  3. 4141