Numford
Vectors

Resolve a vector into rectangular components

10 at 30 degrees splits into Ax = 8.66 along x and Ay = 5 along y.

Vectors

Drag the glowing tips, or type values. Angles are measured from the x axis, and θ is the angle from A to B. Pick a result card to see it built in the picture.

Angle between90°cos θ = A·B / AB
Unit vectors0.87i + 0.5jB = -0.5i + 0.87j

Vector A of size 10 and vector B of size 6 with 90 degrees between them. The resultant is 11.66, 30.96 degrees from A. The dot product is 0 and the cross product has size 60.

Show the formulas
Resultant
R = √(A² + B² + 2AB cos θ) = √(10² + 6² + 2 × 10 × 6 × cos 90°) = 11.66
Direction
tan β = B sin θ / (A + B cos θ) = 6 × sin 90° / (10 + 6 × cos 90°), so β = 30.96° from A
Difference
|A - B| = √(A² + B² - 2AB cos θ) = √(10² + 6² - 2 × 10 × 6 × cos 90°) = 11.66
Dot product
A·B = AB cos θ = 10 × 6 × cos 90° = 0
Cross product
|A×B| = AB sin θ = 10 × 6 × sin 90° = 60
Components
A·B = AxBx + AyBy + AzBz = 8.66 × -3 + 5 × 5.2 + 0 × 0 = 0
Determinant
A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 60k
Unit vector
 = A / |A| = 0.87i + 0.5j
Working

Step by step

  1. Given A = 10 at θ = 30° from the x axis.
  2. Ax = A cos θ = 10 × cos 30° = 8.66
  3. Ay = A sin θ = 10 × sin 30° = 5
  4. A = 8.66i + 5j. Check: √(75 + 25) = 10.
Derivation

Why it works

  1. Choose perpendicular axes and let A make the angle θ with the x axis.
  2. Drop perpendiculars from the tip of A to the two axes to form a right triangle with A as the hypotenuse.
  3. The side along x is adjacent to θ, so Ax = A cos θ. The side along y is opposite, so Ay = A sin θ.
  4. Together they rebuild the vector: A = Ax i + Ay j, with A = √(Ax² + Ay²).
  5. The angle comes back from tan θ = Ay / Ax.
Questions

Rectangular components: quick answers

How do you resolve a vector into components?

For a vector A at angle θ from the x axis, Ax = A cos θ and Ay = A sin θ. The two are at right angles, so they are called rectangular components.

What are the components of 10 at 30 degrees?

Ax = 8.66 and Ay = 5, so A = 8.66i + 5j.

Can a component be larger than the vector?

No. A component is the side of a right triangle whose hypotenuse is the vector, so it is at most the vector's magnitude.

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