Q.Find a vector of magnitude 5 units, and parallel to the resultant of the vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + \hat{k}$.

Concept: A vector parallel to the resultant is a scalar multiple of it. The magnitude condition fixes the scalar.

Step 1: Find the resultant vector.

$\vec{R} = \vec{a} + \vec{b} = (2+1)\hat{i} + (3-2)\hat{j} + (-1+1)\hat{k} = 3\hat{i} + \hat{j} + 0\hat{k}$.

Step 2: Magnitude of $\vec{R}$:

$|\vec{R}| = \sqrt{3^2 + 1^2 + 0^2} = \sqrt{10}$.

Step 3: A unit vector parallel to $\vec{R}$ is $\frac{\vec{R}}{|\vec{R}|} = \frac{3\hat{i} + \hat{j}}{\sqrt{10}}$.

Step 4: Multiply by the desired magnitude 5: …

The core idea here is simple: "parallel to the resultant" means the vector we want points in exactly the same direction as $\vec{a} + \vec{b}$, but can be any length. So we first find the resultant, then scale it to the required magnitude.

Why this works:

If two vectors are parallel, one is a scalar multiple of the other. So if $\vec{r} = \vec{a} + \vec{b}$ is the resultant, any vector parallel to it is $k\vec{r}$ for some real $k$. The magnitude of $k\vec{r}$ is $|k| \cdot |\vec{r}|$. To get a magnitude of exactly 5, we set $|k| = \frac{5}{|\vec{r}|}$. The sign of $k$ just flips the direction — both are parallel, so either is acceptable unless the problem specifies a particular sense.

Let's work through it.

1. Find the resultant vector Add the given vectors component-wise:

Notice the $\hat{k}$ component cancels out — the resultant lies entirely in the $xy$-plane.

2. Find the magnitude of the resultant

3. Find the unit vector in the direction of $\vec{r}$ A unit vector along $\vec{r}$ is:

4. Scale to magnitude 5 Multiply the unit vector by 5: $\vec{v} = 5 \cdot \hat{r} = \frac{15}{\sqrt{10}}\hat{i} + \frac{5}{\sqrt{10}}\hat{j}$ …

Method: A Vector of Given Magnitude Parallel to Another

Use this to build a vector of a specified length pointing along a given (or resultant) direction.

Steps

Step 1: Find the direction vector

If 'parallel to the resultant', first add the given vectors: $\vec r=\vec a+\vec b$ (component-wise). This is the direction to point along.

Step 2: Normalise to a unit vector

$\hat r=\frac{\vec r}{|\vec r|},\qquad |\vec r|=\sqrt{r_x^2+r_y^2+r_z^2}.$ …

Common Mistakes

Mistake 1: Skipping normalisation and just scaling the resultant

Why it's wrong: $m\,\vec r$ has magnitude $m|\vec r|$, not $m$; you must divide by $|\vec r|$ first. Correct approach: use $\dfrac{m}{|\vec r|}\vec r$.

Mistake 2: Inventing a $\hat k$ component that cancelled

Why it's wrong: here $\vec a+\vec b$ has zero $z$-component, so the answer stays in the $xy$-plane; adding a $\hat k$ term is wrong. Correct approach: keep the zero component zero. …