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Vectors: magnitude, direction and products

Resolve parallel and perpendicular components · Diagrams

Projection in acute and obtuse cases

Compare the signed scalar, the parallel vector and the perpendicular residual. The illustration keeps the target unit direction pointing right.

Illustration: |a| = 3 units and b̂ points right. Change θ to compare signs and lengths. Values are rounded to four significant figures.

Angle θ between a and b45°
Try an example
PQRθb̂

Acute angle

The projection follows the target direction; its signed component is positive.

Signed scalar s
2.121
Parallel length PQ
2.121 units
Perpendicular length QR
2.121 units

PR represents a, PQ its parallel vector and QR its perpendicular residual. The angle θ is measured from b̂ to a.

Keep the scalar component's sign

s=a⋅b^s=\mathbf a\cdot\widehat{\mathbf b}

Use the unit direction b^=b/∣b∣\widehat{\mathbf b}=\mathbf b/|\mathbf b|, with b≠0\mathbf b\ne\mathbf0. A negative component means the projection points opposite to this direction.

Resolve the parallel vector

a∥=sb^\mathbf a_\parallel=s\widehat{\mathbf b}

The arrow PQ→\overrightarrow{PQ} follows the target direction when s is positive and reverses when s is negative. Keep the sign in the vector answer.

Find the perpendicular residual

a⊥=a−sb^\mathbf a_\perp=\mathbf a-s\widehat{\mathbf b}

Triangle addition gives PR→=PQ→+QR→\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}. The residual QR→\overrightarrow{QR} is perpendicular to the target direction.

Take a magnitude for the parallel length

d=PQ=∣a⋅b^∣d=PQ=|\mathbf a\cdot\widehat{\mathbf b}|

PQ is a nonnegative length. The same absolute-value formula covers acute and obtuse angles.

The target vector b must be nonzero to form b̂. If a is zero, both lengths are zero and its angle has no defined direction.

Compare with the acute and obtuse source cases

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Two source triangles have PR=a and Q as the perpendicular foot. In the acute case b points from P towards Q; in the obtuse case b points left while PQ points right. The angle between a,b is θ, while the obtuse case triangle angle is π−θ. PQ is the nonnegative length; vector PQ is the parallel component (ii), QR the perpendicular component (iii).
Two source triangles have PR=a and Q as the perpendicular foot. In the acute case b points from P towards Q; in the obtuse case b points left while PQ points right. The angle between a,b is θ, while the obtuse case triangle angle is π−θ. PQ is the nonnegative length; vector PQ is the parallel component (ii), QR the perpendicular component (iii).
Two source triangles have PR=a and Q as the perpendicular foot. In the acute case b points from P towards Q; in the obtuse case b points left while PQ points right. The angle between a,b is θ, while the obtuse case triangle angle is π−θ. PQ is the nonnegative length; vector PQ is the parallel component (ii), QR the perpendicular component (iii).
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