Q.A bird flies through a distance in a straight line given by the vector 𝑖̂ + 2𝑗̂ + 𝑘̂ . A man standing beside a straight metro rail track given by 𝑟⃗ = (3 + λ)𝑖̂ + (2λ − 1)𝑗̂ + 3λ𝑘̂ is observing the bird. The projected length of its flight on the metro track is
(A) 6 √14 units
(B) 14 √6 units
(C) 8 √14 units
(D) 5 √6 units

The projected length of the flight on the track is the scalar projection of the flight vector onto the track's direction.

Vectors. Flight $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$. The track $\vec{r} = (3+\lambda)\hat{i} + (2\lambda-1)\hat{j} + 3\lambda\hat{k}$ has direction $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ (the coefficients of $\lambda$).

Projection. $\dfrac{|\vec{a}\cdot\vec{b}|}{|\vec{b}|}$.

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

The idea

The bird's flight is a vector; the metro track is a straight line with a fixed direction. The "projected length of the flight on the track" is the length of the shadow the flight vector casts along the track, i.e. the scalar projection of the flight vector onto the track's direction vector.

Set up

Flight vector:

The track is $\vec{r} = (3+\lambda)\hat{i} + (2\lambda-1)\hat{j} + 3\lambda\hat{k}$. Splitting the fixed part from the $\lambda$ part,

so the track's direction is

Only the direction matters for a projection; the constant part just fixes where the line sits.

Work the steps

1. Dot product.

2. Length of the direction.

3. Scalar projection. …

Method: Projecting a vector onto a line's direction

Use this whenever you must find how much of one vector lies along a given line — the length of its "shadow" on that line (the projected length).

Steps

Step 1: Extract the line's direction vector.

A line written as $\vec{r} = (\text{fixed part}) + \lambda(\text{direction})$ has direction equal to the coefficients of the parameter $\lambda$ only. The constant part just fixes where the line sits and plays no role in a projection.

Step 2: Write the projected length as a scalar projection.

The length of the shadow of $\vec{a}$ on the direction $\vec{b}$ is

$\text{projected length} = \frac{|\vec{a}\cdot\vec{b}|}{|\vec{b}|}.$ …

Common Mistakes

Mistake 1: Taking the track's direction from the whole expression instead of the $\lambda$-coefficients.

Why it's wrong: only the coefficients of $\lambda$, here $(1,2,3)$, give the line's direction; the constant part $(3,-1,0)$ merely locates the line. Correct approach: project onto $\vec{b}=\hat{i}+2\hat{j}+3\hat{k}$.

Mistake 2: Confusing the scalar projection with the vector projection.

Why it's wrong: the projected LENGTH is $\frac{|\vec{a}\cdot\vec{b}|}{|\vec{b}|}$ (divide by $|\vec{b}|$), whereas dividing by $|\vec{b}|^2$ and multiplying by $\vec{b}$ produces a vector, not a length. Correct approach: use $\frac{|\vec{a}\cdot\vec{b}|}{|\vec{b}|}=\frac{8}{\sqrt{14}}$. …