SME
Grade 12 Math · Precalculus
Matrices as transformations of the plane
Use this for “Rotate (3, 1) by 90° about the origin with a rotation matrix.”
Diagram
Rotate 90° counterclockwise about (0, 0): (x, y) → (−y, x).
A(3, 1) → A′(−1, 3); B′(0, 0).
The image has the same side lengths and angles.
cos θ: 0
sin θ: 1
This is an example. Type your own values to start a new problem.
cos
°
sin
°
−sin
°
cos
°
x
y
=
x′
y′
c
cos θ
worked out (not typed)
s
sin θ
worked out (not typed)
Formulas
c
=
cos(
θ
)
0 = cos(90°)
s
=
sin(
θ
)
1 = sin(90°)
x
′
=
c
×
x
−
s
×
y
−1 = 0 × 3 − 1 × 1
y
′
=
s
×
x
+
c
×
y
3 = 1 × 3 + 0 × 1
x
=
c
×
x
′
+
s
×
y
′
3 = 0 × (−1) + 1 × 3
y
=
−
s
×
x
′
+
c
×
y
′
1 = −1 × (−1) + 0 × 3
Assumptions
•
Turning by θ about the origin is the matrix [[cos θ, −sin θ], [sin θ, cos θ]]; a positive θ turns counterclockwise.
•
Its columns are where (1, 0) and (0, 1) land: (cos θ, sin θ) and (−sin θ, cos θ).
•
Multiply the matrix by the point as a column: each row times the column gives one coordinate.
•
The page turns by an angle you type; it does not work the angle back from a point and its image.
Step-by-step
WE KNOW
θ = 90°, x = 3, y = 1
FIND
cos θ (c), sin θ (s), x of the image (x′), y of the image (y′)
Step 1 · Find cos θ (c)
Use c = cos(θ)
The
first
column
is
where
(1,
0)
lands:
(cos
θ
,
sin
θ
).
c
=
cos(90°)
c
=
0
Step 2 · Find sin θ (s)
Use s = sin(θ)
The
same
column’
s
second
entry.
s
=
sin(90°)
s
=
1
Step 3 · Find x of the image (x′)
Use x′ = c × x − s × y
The
first
row
[cos
θ
,
−sin
θ
]
times
the
column
(
x
,
y
).
x
′
=
0
×
3
−
1
×
1
x
′
=
0
−
1
x
′
=
−1
Step 4 · Find y of the image (y′)
Use y′ = s × x + c × y
The
second
row
[sin
θ
,
cos
θ
]
times
the
column
(
x
,
y
).
y
′
=
1
×
3
+
0
×
1
y
′
=
3
Check
Put the numbers back in. Both sides should match.
0 = cos(90°) ✓
1 = sin(90°) ✓
−1 = 0 × 3 − 1 × 1 ✓
3 = 1 × 3 + 0 × 1 ✓
3 = 0 × (−1) + 1 × 3 ✓
1 = −1 × (−1) + 0 × 3 ✓
Topic check
See what you know: 5 questions on this topic
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More problem types
The image of a point, and undoing it
Use this for “Where does [[2, 1], [1, 1]] send (3, −1)? Which point does it send to (5, 2)?”
How a matrix changes area
Use this for “A shape of area 4 is transformed by [[3, 1], [1, 2]]. What is the new area?”
Two turns in a row
Use this for “Turn (4, 2) by 30° and then by 60°. Which single matrix does both?”
Which move does the matrix make?
Use this for “Which transformation does [[0, 1], [1, 0]] make?”