Q.Projection vector of $\vec{a}$ on $\vec{b}$ is
(A) $\left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}$
(B) $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$
(C) $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|}$
(D) $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|^2}\hat{b}$

Concept: Vector Projection — the projection of $\vec{a}$ onto $\vec{b}$ gives a vector along $\vec{b}$ whose magnitude is the scalar projection $\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$.

Step 1: The scalar projection (component) of $\vec{a}$ along $\vec{b}$ is $\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$.

Step 2: To get the vector projection, multiply this scalar by the unit vector in the direction of $\vec{b}$, which is $\frac{\vec{b}}{|\vec{b}|}$. …

The idea of a projection is simple: if you shine a light straight down onto a line, the shadow a vector casts on that line is its projection. For vectors, the projection of $\vec{a}$ onto $\vec{b}$ answers: "How much of $\vec{a}$ points in the direction of $\vec{b}$, and what is that vector?"

This is not a scalar — it is a vector itself. It has a magnitude (the length of the shadow) and a direction (the direction of $\vec{b}$). So the formula must produce a vector that is parallel to $\vec{b}$.

Let’s build it step by step.

1. Find the scalar component of $\vec{a}$ along $\vec{b}$. The dot product $\vec{a}\cdot\vec{b}$ gives $|\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between them. The quantity $|\vec{a}|\cos\theta$ is the length of the projection of $\vec{a}$ onto the line of $\vec{b}$. To isolate this, divide the dot product by $|\vec{b}|$:

This is the scalar projection (also called the component of $\vec{a}$ along $\vec{b}$). It tells you how long the shadow is, but not the vector itself.

2. Turn that scalar into a vector. To get the actual projection vector, we need to multiply this scalar length by a unit vector in the direction of $\vec{b}$. The unit vector along $\vec{b}$ is $\hat{b} = \frac{\vec{b}}{|\vec{b}|}$. So:

3. Match with the options. Option (A) is exactly $\left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}$. Option (B) is the scalar projection (missing the direction). …

Method: Building the vector projection of one vector onto another

Use this whenever you need the vector (not just scalar) projection of $\vec{a}$ onto $\vec{b}$ — the "shadow" of $\vec{a}$ along $\vec{b}$.

Steps

Step 1: Get the scalar projection (the length of the shadow)

This is a signed number: how much of $\vec a$ points along $\vec b$.

Step 2: Give it a direction with the unit vector along $\vec{b}$

Multiply the scalar by $\hat{b}=\dfrac{\vec{b}}{|\vec{b}|}$ so the result points along $\vec b$:

$\text{proj}_{\vec b}\,\vec a=\left(\frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}\right)\frac{\vec{b}}{|\vec{b}|}$ …

Common Mistakes

Mistake 1: Confusing the scalar projection with the vector projection

Why it's wrong: $\dfrac{\vec a\cdot\vec b}{|\vec b|}$ is only a number (the shadow's length); the vector projection must also carry $\vec b$'s direction. Correct approach: multiply by $\hat b$ to get $\left(\dfrac{\vec a\cdot\vec b}{|\vec b|^2}\right)\vec b$.

Mistake 2: Using $|\vec b|$ instead of $|\vec b|^2$ in the denominator

Why it's wrong: you normalise once to form $\hat b$ and once for the scalar component, giving $|\vec b|^2$. A single power leaves the length wrong. Correct approach: denominator is $|\vec b|^2$. …