SME
Grade 12 Math · Precalculus
Vectors: components, magnitude and dot product
Use this for “A vector has magnitude 10 at 30°. Find its components,” or the magnitude and direction from components.
Diagram
v: 10 at 30°, so v = ⟨8.66, 5⟩.
x-component: 8.6603
y-component: 5
This is an example. Type your own values to start a new problem.
vₓ
=
|v|
cos
θ
°
vᵧ
=
|v|
sin
θ
°
Formulas
vₓ
=
|v|
×
cos(
θ
°)
8.6603 = 10 × cos(30°)
vᵧ
=
|v|
×
sin(
θ
°)
5 = 10 × sin(30°)
|v|
=
√
vₓ² + vᵧ²
10
=
√
8.6603
2
+
5
2
tan
θ
=
vᵧ
vₓ
,
θ
in
the
quadrant
of
(vₓ,
vᵧ)
tan
30
=
5
8.6603
,
30
in
the
quadrant
of
(8.6603,
5)
Assumptions
•
The direction θ is measured counterclockwise from the positive x-axis.
•
A vector has a magnitude (its length) and a direction but no fixed place: move it and it is the same vector.
•
tan⁻¹(vᵧ ÷ vₓ) alone gives the wrong quadrant when vₓ < 0: add 180°.
Step-by-step
WE KNOW
|v| = 10, θ = 30°
FIND
x-component (vₓ), y-component (vᵧ)
Step 1 · Find x-component (vₓ)
Use vₓ = |v| × cos(θ°)
The x-component is the magnitude times the cosine of the direction.
vₓ = 10 × cos(30°)
vₓ = 10 × 0.866
vₓ = 8.6603
Step 2 · Find y-component (vᵧ)
Use vᵧ = |v| × sin(θ°)
The y-component is the magnitude times the sine of the direction.
vᵧ = 10 × sin(30°)
vᵧ = 10 × 0.5
vᵧ = 5
Check
Put the numbers back in. Both sides should match.
8.6603 ≈ 10 × cos(30°) ✓
5 = 10 × sin(30°) ✓
10
≈
√
8.6603
2
+
5
2
✓
tan(30°)
=
5
8.6603
✓
Topic check
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