Skip to content

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 NCERT 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. …

Start your 14-day free trial to unlock the full solution →
← Previous · 10.6.1Scalar (or Dot) Product of Two VectorsNext · 10.6.2 →Example 13 — Find the angle between two vectors veca and vecb with magnitudes 1 and…