x-Component from Magnitude and Angle
Worked example: Magnitude 10 at 60° → v_x = 5 — press Try an example to run it live, then adjust anything.
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Casting the shadows →
Grade 12Grade 12 Math
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x-Component from Magnitude and Angle explained
Resolving a vector is the reverse of building one: the shadow it casts on the x-axis has length |v| cos θ. Simon Stevin demonstrated the underlying parallelogram rule in 1586 with an inclined-plane thought experiment — a closed loop of beads draped over a wedge that would have to move forever if forces did not combine this way — and Newton restated it as Corollary I of the Principia. Every ramp problem since is an application: a 200 N pull on a sled at 25° above the ground drives the sled forward with 200 × cos 25° ≈ 181 N, while the rest of the effort merely lifts.
The classic trap is measuring the angle from the wrong line. The cosine belongs to the component along the axis the angle is measured from; take the angle from the vertical instead and cosine and sine swap places. Check by sanity: at θ = 0° the whole vector lies on x, and cos 0° = 1 delivers exactly that.
x-Component from Magnitude and Angle formula
- = x-component
- = Vector magnitude
- = Angle from +x axis (°)
Missing one of these? Work it out first, then come back
- x-component — y-Component from Magnitude and Angle, Direction Angle of a 2D Vector
- Vector magnitude — y-Component from Magnitude and Angle, Unit Vector Component (Normalization)
- Angle from +x axis — y-Component from Magnitude and Angle, Direction Angle of a 2D Vector