SME
Grade 12 Math · Problem type
Adding vectors
Use this for “Find u + v for u = ⟨3, 1⟩ and v = ⟨1, 2⟩, and its magnitude.”
Diagram
u = ⟨3, 1⟩
|u|
=
√
10
≈
3.16.
v = ⟨1, 2⟩
|v|
=
√
5
≈
2.24.
Tip to tail: u + v = ⟨4, 3⟩
|u + v| = 5 at 36.87°.
This is an example. Type your own values to start a new problem.
⟨
u₁
,
u₂
⟩
+
⟨
v₁
,
v₂
⟩
=
⟨
s₁
,
s₂
⟩
|s|
Magnitude of s = u + v
calculated
Formulas
s₁
=
u₁
+
v₁
4 = 3 + 1
s₂
=
u₂
+
v₂
3 = 1 + 2
|s|
=
√
s₁
2
+
s₂
2
5
=
√
4
2
+
3
2
Assumptions
•
Add vectors by adding matching components.
•
Tip to tail: start v where u ends; u + v runs from u’s tail to v’s tip.
•
The magnitude of the sum is not the sum of the magnitudes.
Step-by-step
WE KNOW
u₁ = 3, u₂ = 1, v₁ = 1, v₂ = 2
FIND
x-component of u + v (s₁), y-component of u + v (s₂), magnitude of s = u + v (|s|)
Step 1 · Find x-component of u + v (s₁)
Use s₁ = u₁ + v₁
Add the x-components.
s₁
=
3
+
1
s₁
=
4
Step 2 · Find y-component of u + v (s₂)
Use s₂ = u₂ + v₂
Add the y-components.
s₂
=
1
+
2
s₂
=
3
Step 3 · Find magnitude of s = u + v (|s|)
Use |s| = √(s₁² + s₂²)
The Pythagorean theorem on the sum’s components.
|s|
=
√
4
2
+
3
2
|s|
=
√
16 + 9
|s|
=
√
25
|s| = 5
Check
Put the numbers back in. Both sides should match.
4 = 3 + 1 ✓
3 = 1 + 2 ✓
5
=
√
4
2
+
3
2
✓
Topic check
See what you know: 5 questions on this topic
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