Q.Find a vector in the direction of vector $5\hat{i} - \hat{j} + 2\hat{k}$ which has magnitude 8 units.

Concept: Unit Vector Scaling — any vector in a given direction is a scalar multiple of the unit vector along that direction.

Step 1: Find the magnitude of the given vector $\vec{a} = 5\hat{i} - \hat{j} + 2\hat{k}$.

Step 2: The unit vector in the direction of $\vec{a}$ is:

$\hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{5\hat{i} - \hat{j} + 2\hat{k}}{\sqrt{30}}$ …

Concept and Intuition: Unit Vector Scaling

A vector has two independent properties: direction and magnitude. If you want a vector that points exactly the same way as a given vector $\vec{a}$ but has a different length, you first strip away the original length to get a pure direction (a unit vector), then stretch that direction to the new length.

Mathematically, the unit vector in the direction of $\vec{a}$ is $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$. This vector has magnitude 1 and points exactly along $\vec{a}$. To get a vector of magnitude $m$ in that same direction, you simply multiply: $\vec{v} = m \cdot \hat{a}$.

Here, $\vec{a} = 5\hat{i} - \hat{j} + 2\hat{k}$ and $m = 8$.

Step-by-Step Solution

1. Find the magnitude of the given vector

The magnitude of $\vec{a} = 5\hat{i} - \hat{j} + 2\hat{k}$ is:

2. Construct the unit vector in the same direction

The unit vector $\hat{a}$ is obtained by dividing each component of $\vec{a}$ by its magnitude:

This vector has magnitude exactly 1. You can verify:

3. Scale the unit vector to the desired magnitude

We want magnitude 8, so multiply $\hat{a}$ by 8:

Distribute the 8: …

Method: Building a Vector of a Required Magnitude in a Given Direction

Use this whenever you need a vector that points along a given vector $\vec{a}$ but has a specified length $m$ (here $m=8$). The idea: separate direction from length, then rebuild.

Steps

Step 1: Find the direction as a unit vector.

This strips $\vec{a}$ down to pure direction (length $1$).

Step 2: Scale the unit vector up to the required magnitude.

Because $|m\hat{a}|=|m|\,|\hat{a}|=m\cdot1=m$, the result automatically has the length you want. …

Common Mistakes

Mistake 1: Multiplying the original vector by the required magnitude directly.

Why it's wrong: $8(5\hat{i}-\hat{j}+2\hat{k})$ has magnitude $8|\vec{a}|=8\sqrt{30}$, not $8$. You must normalise first. Correct approach: divide by $|\vec{a}|$ to get length $1$, then multiply by $8$.

Mistake 2: Sign error when squaring $-1$.

Why it's wrong: $|\vec{a}|=\sqrt{5^2+(-1)^2+2^2}=\sqrt{30}$; $(-1)^2=+1$, not $-1$. Correct approach: squaring always gives a non-negative result. …