SME
Calculus III (Multivariable)
Vectors and 3D geometry
Use this for “Find the plane through (1, 0, 3) with normal ⟨2, −1, 2⟩, and the distance from (3, 1, 4) to it.”
Diagram
n = ⟨2, −1, 2⟩ through P₀(1, 0, 3): d = 2 × 1 − 1 × 0 + 2 × 3 = 8, so 2x − y + 2z = 8.
Q(3, 1, 4) drops along n to F(1.889, 1.556, 2.889): D = |2 × 3 − 1 × 1 + 2 × 4 − 8| ÷ 3 = 1.667.
This is an example. Type your own values to start a new problem.
n
=
⟨
a
,
b
,
c
⟩,
P₀
=
(
x₀
,
y₀
,
z₀
)
a
x
+
(
b
)
y
+
c
z
=
d
Q
=
(
Q₁
,
Q₂
,
Q₃
),
D
=
5
3
s
Plane’s left side at Q minus d (n · Q − d)
worked out (not typed)
|n|²
Length of n squared
worked out (not typed)
F₁
Foot F, x
worked out (not typed)
F₂
Foot F, y
worked out (not typed)
F₃
Foot F, z
worked out (not typed)
Formulas
d
=
a
×
x₀
+
b
×
y₀
+
c
×
z₀
8 = 2 × 1 + (−1) × 0 + 2 × 3
s
=
a
×
Q₁
+
b
×
Q₂
+
c
×
Q₃
−
d
5 = 2 × 3 + (−1) × 1 + 2 × 4 − 8
|n|²
=
a
2
+
b
2
+
c
2
9
=
2
2
+
(−1)
2
+
2
2
D
=
|
s
|
÷
√|n|²
5
3
=
|5|
÷
√
9
F₁
=
Q₁
−
a
×
s
÷
|n|²
17
9
=
3
−
2
×
5
9
F₂
=
Q₂
−
b
×
s
÷
|n|²
14
9
=
1
−
(−1)
×
5
9
F₃
=
Q₃
−
c
×
s
÷
|n|²
26
9
=
4
−
2
×
5
9
Assumptions
•
Every vector from P₀ to a point of the plane is square to n, so n · (r − P₀) = 0.
•
Written out, the plane is ax + by + cz = d with d = n · P₀.
•
The distance D is measured along n, from Q straight to its foot F on the plane.
Step-by-step
WE KNOW
a = 2, b = −1, c = 2, x₀ = 1, y₀ = 0, z₀ = 3, Q₁ = 3, Q₂ = 1, Q₃ = 4
FIND
plane constant (d), plane’s left side at Q minus d (n · Q − d) (s), length of n squared (|n|²), distance from Q (D), foot F, x (F₁), foot F, y (F₂), foot F, z (F₃)
Step 1 · Find plane constant (d)
Use d = a × x₀ + b × y₀ + c × z₀
n
·
(r
−
P₀)
=
0
means
n
·
r
=
n
·
P₀:
the
dot
product
of
n
with
P₀
is
d
.
d
=
2
×
1
+
(−1)
×
0
+
2
×
3
d
=
2
+
6
d
=
8
→
The
plane:
2x
−
y
+
2z
=
8
Step 2 · Find plane’s left side at Q minus d (n · Q − d): s
Use s = a × Q₁ + b × Q₂ + c × Q₃ − d
Put
Q
into
the
left
side
ax
+
by
+
cz
and
take
d
away:
0
would
mean
Q
is
on
the
plane.
s
=
2
×
3
+
(−1)
×
1
+
2
×
4
−
8
s
=
6
+
(−1)
+
8
−
8
s
=
5
Step 3 · Find length of n squared (|n|²)
Use |n|² = a² + b² + c²
The squared length of n, used by the distance and by the foot.
|n|²
=
2
2
+
(−1)
2
+
2
2
|n|² = 4 + 1 + 4
|n|² = 9
Step 4 · Find distance from Q (D)
Use D = |s| ÷ √|n|²
n
·
Q
−
d
is
how
far
off
the
plane
Q
is,
times
|n|,
so
divide
by
|n|.
D
=
|5|
÷
√
9
D
=
5
3
D
=
5
3
Step 5 · Find foot F, x (F₁)
Use F₁ = Q₁ − a × s ÷ |n|²
Step
back
from
Q
along
n
by
(n
·
Q
−
d
)
÷
|n|²
of
n,
which
lands
on
the
plane.
F₁
=
3
−
2
×
5
9
F₁
=
3
−
10
9
F₁
=
3
−
1.111
F₁
=
17
9
Step 6 · Find foot F, y (F₂)
Use F₂ = Q₂ − b × s ÷ |n|²
Step
back
from
Q
along
n
by
(n
·
Q
−
d
)
÷
|n|²
of
n,
which
lands
on
the
plane.
F₂
=
1
−
(−1)
×
5
9
F₂
=
1
−
−5
9
F₂
=
1
−
(−0.5556)
F₂
=
14
9
Step 7 · Find foot F, z (F₃)
Use F₃ = Q₃ − c × s ÷ |n|²
Step
back
from
Q
along
n
by
(n
·
Q
−
d
)
÷
|n|²
of
n,
which
lands
on
the
plane.
F₃
=
4
−
2
×
5
9
F₃
=
4
−
10
9
F₃
=
4
−
1.111
F₃
=
26
9
Check
Put the numbers back in. Both sides should match.
8 = 2 × 1 + (−1) × 0 + 2 × 3 ✓
5 = 2 × 3 + (−1) × 1 + 2 × 4 − 8 ✓
9
=
2
2
+
(−1)
2
+
2
2
✓
5
3
=
|5|
÷
√
9
✓
17
9
=
3
−
2
×
5
9
✓
14
9
=
1
−
(−1)
×
5
9
✓
26
9
=
4
−
2
×
5
9
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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