Sample 2025-26 Class 12 Maths Q13 Solution: Projected length of a bird's flight on a metro track

The question

A bird flies through a distance in a straight line given by the vector $\hat i + 2\hat j + \hat k$ ⁠. A man standing beside a straight metro rail track given by $\vec r = (3 + \lambda)\hat i + (2\lambda - 1)\hat j + 3\lambda\hat k$ is observing the bird. The projected length of its flight on the metro track is

(A) $\frac{6}{\sqrt{14}}$ units

(B) $\frac{14}{\sqrt{6}}$ units

(C) $\frac{8}{\sqrt{14}}$ units

(D) $\frac{5}{\sqrt{6}}$ units

Exam answer

The track is $\vec r = (3\hat i - \hat j) + \lambda\vec b$ ⁠, where $\vec b = \hat i + 2\hat j + 3\hat k$ ⁠.

Why?

Collect the terms without $\lambda$ and the terms with $\lambda$ ⁠:

\begin{aligned} \vec r &= (3\hat i - \hat j) \\ &\quad + \lambda(\hat i + 2\hat j + 3\hat k) \end{aligned}

This is the form $\vec r = \vec a_1 + \lambda\vec b$ ⁠. The vector multiplying $\lambda$ is the direction of the track.

Projection of $\vec a = \hat i + 2\hat j + \hat k$ on $\vec b$ ⁠:

\begin{aligned} &\frac{\vec a \cdot \vec b}{|\vec b|} \\ &= \frac{1 \times 1 + 2 \times 2 + 1 \times 3}{\sqrt{1^2 + 2^2 + 3^2}} \\ &= \frac{8}{\sqrt{14}} \text{ units} \end{aligned}

Why?

The projected length on the track is the projection of the flight vector on the track's direction:

\text{projection} = \frac{\vec a \cdot \vec b}{|\vec b|}

$\vec a \cdot \vec b = 1 + 4 + 3 = 8$ and $|\vec b| = \sqrt{1 + 4 + 9} = \sqrt{14}$ ⁠.

Watch out: divide by $|\vec b|$ ⁠, not by $|\vec a| = \sqrt{6}$ as in (B) and (D).

Answer: (C) $\dfrac{8}{\sqrt{14}}$ units

Where this comes from

Chapter 10, Vector Algebra: Projection of a vector on a line.

Question type: Projection

How to spot it: click to see

Type 6. Projection: the shadow of one vector on another

Spot it: "find the projection of the vector $\vec a$ on the vector $\vec b$ ".

Here: The track's direction is the vector beside $\lambda$ ⁠, $\hat i + 2\hat j + 3\hat k$ ⁠. Project the flight $\hat i + 2\hat j + \hat k$ on it: dot product 8, divided by $\sqrt{14}$ ⁠.

Quick lesson · Worked NCERT solutions · Interactive lesson

Also uses Chapter 11, Three Dimensional Geometry.

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Q13: Projected length of a bird's flight on a metro track