SME
Calculus III (Multivariable) · Problem type
A line and where it meets a plane
Use this for “Where does the line through (1, 2, −1) along ⟨2, 0, 3⟩ meet the plane x + y + z = 10?”
Diagram
r(t) = (1, 2, −1) + t⟨2, 0, 3⟩.
On x + y + z = 10: 2 + 5t = 10, so t = 1.6.
This is an example. Type your own values to start a new problem.
r(t)
=
(
x₀
,
y₀
,
z₀
)
+
t⟨
v₁
,
v₂
,
v₃
⟩
a
x
+
b
y
+
c
z
=
d
r(
t
)
=
(
x
,
y
,
z
)
n · P₀
Plane’s left side at P₀
worked out (not typed)
n · v
Change per unit of t
worked out (not typed)
Formulas
n
·
P₀
=
a
×
x₀
+
b
×
y₀
+
c
×
z₀
2 = 1 × 1 + 1 × 2 + 1 × (−1)
n
·
v
=
a
×
v₁
+
b
×
v₂
+
c
×
v₃
5 = 1 × 2 + 1 × 0 + 1 × 3
t
=
d
− (n · P₀)
n · v
1.6
=
10 − 2
5
x
=
x₀
+
t
×
v₁
4.2 = 1 + 1.6 × 2
y
=
y₀
+
t
×
v₂
2 = 2 + 1.6 × 0
z
=
z₀
+
t
×
v₃
3.8 = −1 + 1.6 × 3
Assumptions
•
r(t) = P₀ + tv runs through P₀ along the direction v; t = 0 is P₀ itself.
•
The line meets the plane where r(t) satisfies ax + by + cz = d, one value of t.
•
If n · v = 0 the line runs parallel to the plane: it misses it or lies in it.
Step-by-step
WE KNOW
x₀ = 1, y₀ = 2, z₀ = −1, v₁ = 2, v₂ = 0, v₃ = 3, a = 1, b = 1, c = 1, d = 10
FIND
plane’s left side at P₀ (n · P₀), change per unit of t (n · v), parameter where they meet (t), meeting point, x (x), meeting point, y (y), meeting point, z (z)
Step 1 · Find plane’s left side at P₀ (n · P₀)
Use n · P₀ = a × x₀ + b × y₀ + c × z₀
Put P₀ into the left side of the plane’s equation: where the line starts.
n · P₀ = 1 × 1 + 1 × 2 + 1 × (−1)
n · P₀ = 3 + (−1)
n · P₀ = 2
Step 2 · Find change per unit of t (n · v)
Use n · v = a × v₁ + b × v₂ + c × v₃
Each
unit
of
t
adds
n
·
v
to
the
left
side,
so
along
the
line
it
reads
n
·
P₀
+
t
(n
·
v).
n · v = 1 × 2 + 1 × 0 + 1 × 3
n · v = 5
Step 3 · Find parameter where they meet (t)
Use t = (d − (n · P₀)) ÷ (n · v)
Solve
n
·
P₀
+
t
(n
·
v)
=
d
for
t
.
t
=
10 − 2
5
t
=
8
5
t
=
1.6
Step 4 · Find meeting point, x (x)
Use x = x₀ + t × v₁
Walk
t
steps
of
v
from
P₀:
the
point
r(
t
)
on
the
line.
x
=
1
+
1.6
×
2
x
=
1
+
3.2
x
=
4.2
Step 5 · Find meeting point, y (y)
Use y = y₀ + t × v₂
Walk
t
steps
of
v
from
P₀:
the
point
r(
t
)
on
the
line.
y
=
2
+
1.6
×
0
y
=
2
Step 6 · Find meeting point, z (z)
Use z = z₀ + t × v₃
Walk
t
steps
of
v
from
P₀:
the
point
r(
t
)
on
the
line.
z
=
−1
+
1.6
×
3
z
=
−1
+
4.8
z
=
3.8
Check
Put the numbers back in. Both sides should match.
2 = 1 × 1 + 1 × 2 + 1 × (−1) ✓
5 = 1 × 2 + 1 × 0 + 1 × 3 ✓
1.6
=
10 − 2
5
✓
4.2 = 1 + 1.6 × 2 ✓
2 = 2 + 1.6 × 0 ✓
3.8 = −1 + 1.6 × 3 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.