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Mathematics · Ch 10 — Vector Algebra

Projection of a Vector on a Line

10.6.2

Projection of a Vector on a Line

10.6.2 Projection of a Vector on a Line

The Core Idea

When a vector a⃗\vec{a} makes an angle θ\theta with a directed line ll (measured anticlockwise), the projection vector of a⃗\vec{a} on ll is a vector p⃗\vec{p} whose magnitude is ∣a⃗∣∣cos⁡θ∣|\vec{a}||\cos\theta|, with the same direction as ll when cos⁡θ>0\cos\theta > 0 and opposite when cos⁡θ<0\cos\theta < 0. Its magnitude ∣p⃗∣|\vec{p}| is called the projection of a⃗\vec{a} on ll.

Note

The textbook uses "projection" for the scalar (signed) magnitude, and "projection vector" for the full vector carrying both magnitude and direction.

Observations and Key Results

Observation 1: Projection Using a Unit Vector

If p^\hat{p} is the unit vector along a line ll, the projection of a⃗\vec{a} on ll is:

a⃗⋅p^\vec{a} \cdot \hat{p}

because a⃗⋅p^=∣a⃗∣∣p^∣cos⁡θ=∣a⃗∣cos⁡θ\vec{a} \cdot \hat{p} = |\vec{a}||\hat{p}|\cos\theta = |\vec{a}|\cos\theta, the signed magnitude of the projection.

Observation 2: Projection of One Vector on Another

The projection of a⃗\vec{a} on another vector b⃗\vec{b} is:

a⃗⋅b^ora⃗⋅b⃗∣b⃗∣\vec{a} \cdot \hat{b} \quad \text{or} \quad \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

where b^=b⃗∣b⃗∣\hat{b} = \frac{\vec{b}}{|\vec{b}|} is the unit vector in the direction of b⃗\vec{b}.

Projection of a⃗\vec{a} on b⃗\vec{b}

projb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Observation 3: Special Angles

  • If θ=0\theta = 0, then cos⁡θ=1\cos\theta = 1, so the projection vector of a⃗\vec{a} is a⃗\vec{a} itself.
  • If θ=π\theta = \pi, then cos⁡θ=−1\cos\theta = -1, so the projection vector of a⃗\vec{a} is −a⃗-\vec{a} (opposite direction).

Observation 4: Perpendicular Case

If θ=π2\theta = \frac{\pi}{2} or θ=3π2\theta = \frac{3\pi}{2}, then cos⁡θ=0\cos\theta = 0, so the projection vector of a⃗\vec{a} is the zero vector.

Direction Cosines and Projections on Axes

If α\alpha, β\beta, γ\gamma are the direction angles of a⃗=a1i^+a2j^+a3k^\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}, its direction cosines are: …

Figure 10.20Projection of vector AB onto a directed line l shown in four cases as the angle theta ranges over 0-90, 90-180, 180-270 and 270-360 degrees, with foot C and projection vector p in each case.
Fig. 10.20 — Projection of vector AB onto a directed line l shown in four cases as the angle theta ranges over 0-90, 90-180, 180-270 and 270-360 degrees, with foot C and projection vector p in each case.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

The figure is a set of four panels, labelled (i) through (iv), each showing the same basic geometry: a horizontal directed line ll (with an arrowhead indicating its positive direction), a vector AB→\overrightarrow{AB} starting from a point AA on ll, and a dashed perpendicular dropped from BB to meet ll at a foot CC. The segment ACAC along ll is the projection vector p→\overrightarrow{p}.

What changes from panel to panel is the angle θ\theta that AB→\overrightarrow{AB} makes with ll, measured anticlockwise from ll to the vector. The four panels cover the four quadrants of the angle:

  • (i) 0∘<θ<90∘0^\circ < \theta < 90^\circ — the projection p→\overrightarrow{p} points to the right (same direction as ll).
  • (ii) 90∘<θ<180∘90^\circ < \theta < 180^\circ — the projection p→\overrightarrow{p} points to the left (opposite to ll).
  • (iii) 180∘<θ<270∘180^\circ < \theta < 270^\circ — again p→\overrightarrow{p} points left.
  • (iv) 270∘<θ<360∘270^\circ < \theta < 360^\circ — p→\overrightarrow{p} points right, and here BB lies below the line ll.

The core idea is that the projection of a vector onto a directed line is itself a vector — its magnitude is ∣AB→∣ ∣cos⁡θ∣|\overrightarrow{AB}|\,|\cos\theta|, and its direction is the same as ll when cos⁡θ>0\cos\theta > 0 and opposite when cos⁡θ<0\cos\theta < 0. The sign of cos⁡θ\cos\theta alone decides the direction of the projection vector along ll.

Projection vector of a⃗ on line l=(a⃗⋅p^) p^\text{Projection vector of } \vec{a} \text{ on line } l = (\vec{a}\cdot\hat{p})\,\hat{p}

where p^\hat{p} is the unit vector along ll. The scalar a⃗⋅p^\vec{a}\cdot\hat{p} is called the projection (a signed scalar), and its absolute value is the length of the projection vector.

The figure makes this sign convention visual: in panels (i) and (iv), cos⁡θ\cos\theta is positive, so the projection vector points rightward along ll; in (ii) and (iii), cos⁡θ\cos\theta is negative, so it points leftward. The dashed perpendicular from BB to CC is the geometric construction that gives AC=∣AB→∣cos⁡θAC = |\overrightarrow{AB}|\cos\theta — the signed length of the projection.

Watch out

A common mistake is to think the projection is always the length of the shadow on the line. The projection vector has a direction: it points along ll if the angle is acute, and opposite to ll if the angle is obtuse. The scalar projection a⃗⋅p^\vec{a}\cdot\hat{p} can be negative; the magnitude ∣a⃗⋅p^∣|\vec{a}\cdot\hat{p}| is the length. …

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