SME
Linear Algebra · Problem type
A rotation then a reflection
Use this for “Rotate (3, 1) by 90° and then reflect it in the x-axis. Find the image and the one matrix that does both.”
Diagram
Rotate 90° counterclockwise about (0, 0), then reflect across the x-axis.
A(3, 1) → A′(−1, 3) → A″(−1, −3).
Both moves keep lengths and angles, so the final image is congruent to the figure.
This is an example. Type your own values to start a new problem.
FR
=
FR
=
θ
Rotation angle
example · °
c
Cosine of the angle, cos θ
worked out (not typed)
s
Sine of the angle, sin θ
worked out (not typed)
y₁
Rotated point Rx, first coordinate
worked out (not typed)
y₂
Rotated point Rx, second coordinate
worked out (not typed)
Formulas
c
=
cos(
θ
)
0 = cos(90°)
s
=
sin(
θ
)
1 = sin(90°)
y₁
=
c
×
x₁
−
s
×
x₂
−1 = 0 × 3 − 1 × 1
y₂
=
s
×
x₁
+
c
×
x₂
3 = 1 × 3 + 0 × 1
z₁
=
y₁
−1 = −1
z₂
=
−
y₂
−3 = −3
m₁₁
=
c
0 = 0
m₁₂
=
−
s
−1 = −1
m₂₁
=
−
s
−1 = −1
m₂₂
=
−
c
0 = 0
Assumptions
•
R = [[cos θ, −sin θ], [sin θ, cos θ]] turns the plane by θ counterclockwise about 0; F = [[1, 0], [0, −1]] reflects it in the x-axis.
•
R first and then F is the product FR, the later move written on the left. The other order, RF, is usually different.
•
det FR = −1, so FR is one reflection: in the line through 0 at the angle −θ/2.
Step-by-step
WE KNOW
θ = 90°, x₁ = 3, x₂ = 1
FIND
cosine of the angle, cos θ (c), sine of the angle, sin θ (s), rotated point Rx, first coordinate (y₁), rotated point Rx, second coordinate (y₂), final image FRx, first coordinate (z₁), final image FRx, second coordinate (z₂), combined matrix FR, row 1, column 1 (m₁₁), combined matrix FR, row 1, column 2 (m₁₂), combined matrix FR, row 2, column 1 (m₂₁), combined matrix FR, row 2, column 2 (m₂₂)
Step 1 · Find cosine of the angle, cos θ (c)
Use c = cos(θ)
The
first
column
of
R
is
where
(1,
0)
lands
after
the
turn:
(cos
θ
,
sin
θ
).
c
=
cos(90°)
c
=
0
Step 2 · Find sine of the angle, sin θ (s)
Use s = sin(θ)
The second entry of the same column.
s
=
sin(90°)
s
=
1
Step 3 · Find rotated point Rx, first coordinate (y₁)
Use y₁ = c × x₁ − s × x₂
Row
1
of
R,
[cos
θ
,
−sin
θ
],
times
the
column
x.
y₁
=
0
×
3
−
1
×
1
R = [[0, −1], [1, 0]]
Rx = ⟨0 × 3 − 1 × 1, 1 × 3 + 0 × 1⟩ = ⟨−1, 3⟩
y₁
=
−1
Step 4 · Find rotated point Rx, second coordinate (y₂)
Use y₂ = s × x₁ + c × x₂
Row
2
of
R,
[sin
θ
,
cos
θ
],
times
the
column
x.
y₂
=
1
×
3
+
0
×
1
y₂
=
3
Step 5 · Find final image FRx, first coordinate (z₁)
Use z₁ = y₁
F = [[1, 0], [0, −1]] keeps the first coordinate: the x-axis is the mirror.
z₁
=
−1
Step 6 · Find final image FRx, second coordinate (z₂)
Use z₂ = −y₂
F changes the sign of the second coordinate: the point flips to the other side of the x-axis.
z₂
=
−3
Step 7 · Find combined matrix FR, row 1, column 1 (m₁₁)
Use m₁₁ = c
Row
1
of
F,
[1,
0],
keeps
row
1
of
R:
m₁₁
=
cos
θ
.
m₁₁
=
0
Step 8 · Find combined matrix FR, row 1, column 2 (m₁₂)
Use m₁₂ = −s
Row
1
of
FR
is
row
1
of
R,
so
m₁₂
=
−sin
θ
.
m₁₂
=
−1
Step 9 · Find combined matrix FR, row 2, column 1 (m₂₁)
Use m₂₁ = −s
Row 2 of F, [0, −1], gives minus row 2 of R.
m₂₁
=
−1
Step 10 · Find combined matrix FR, row 2, column 2 (m₂₂)
Use m₂₂ = −c
Minus
row
2
of
R
again:
m₂₂
=
−cos
θ
.
[[1, 0], [0, −1]] × [[0, −1], [1, 0]] = [[1 × 0 + 0 × 1, 1 × (−1) + 0 × 0], [0 × 0 + (−1) × 1, 0 × (−1) + (−1) × 0]] = [[0, −1], [−1, 0]]
FRx = ⟨−1, −3⟩, the same as F(Rx)
det
FR
=
−cos²
θ
−
sin²
θ
=
−1:
the
reflection
in
the
line
through
0
at
−45°
m₂₂
=
0
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 ✓
−1 = −1 ✓
−3 = −3 ✓
0 = 0 ✓
−1 = −1 ✓
−1 = −1 ✓
0 = 0 ✓
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