Q.Write two different vectors having same direction.

The key idea is that direction vectors are determined by the ratios of their components, not by their magnitude. Two vectors have the same direction if one is a positive scalar multiple of the other.

Step 1: Choose any non-zero vector, say $\vec{a} = 2\hat{i} + 3\hat{j}$.

Step 2: Multiply $\vec{a}$ by any positive scalar $k$ (e.g., $k = 2$) to get a second vector with the same direction:

Step 3: Verify that the direction ratios are proportional: $\frac{2}{4} = \frac{3}{6} = \frac{1}{2}$, confirming they are parallel and point the same way.

Why This Works: The Concept of Direction Vectors

A vector’s direction is determined by the line it lies along and the sense (which way it points along that line). Two vectors share the same direction if they are parallel and point the same way — not opposite. Mathematically, this means one vector is a positive scalar multiple of the other.

For example, if you have a vector $\vec{a}$, then $k\vec{a}$ for any $k > 0$ points exactly in the same direction as $\vec{a}$. The magnitude changes, but the direction stays identical. This is the simplest way to generate infinitely many vectors with the same direction.

Step-by-Step Construction

1. Pick a base vector.

   Choose any vector — say $\vec{a} = (1, 2)$. This will be our reference direction.

2. Multiply by a positive scalar.

   Take $k = 2$. Then $\vec{b} = 2\vec{a} = (2, 4)$.

   Since $2 > 0$, $\vec{b}$ points exactly along the same line and in the same sense as $\vec{a}$.

3. Verify the direction.

   Compute the unit vectors:

   • For $\vec{a}$: $\frac{(1,2)}{\sqrt{1^2+2^2}} = \left(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right)$.
   • For $\vec{b}$: $\frac{(2,4)}{\sqrt{2^2+4^2}} = \frac{(2,4)}{\sqrt{20}} = \frac{(2,4)}{2\sqrt{5}} = \left(\frac{1}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right)$. They are identical, confirming the same direction.

4. Write the answer.

   Any two vectors of the form $\vec{v}$ and $k\vec{v}$ with $k > 0$ work. A simple pair is $(1,2)$ and $(2,4)$.

Method: Constructing two different vectors with the same direction

Use this "give an example" task with the rule that same direction means a positive scalar multiple.

Steps

Step 1: Recall the same-direction condition

$\vec{a}$ and $\vec{b}$ point the same way when $\vec{b}=k\,\vec{a}$ for some scalar $k>0$. (A negative $k$ reverses the direction.)

Step 2: Pick a base vector and scale by a positive number

Choose any non-zero $\vec{a}$, then multiply by a positive $k\neq1$ to get a different vector of the same direction, e.g. $\vec{a}=2\hat{i}+3\hat{j}$ and $\vec{b}=2\vec{a}=4\hat{i}+6\hat{j}$.

Step 3: Verify via equal unit vectors (or proportional components)

Same direction $\iff$ same unit vector $\dfrac{\vec{v}}{|\vec{v}|}$, equivalently components in the same positive ratio: $\tfrac{2}{4}=\tfrac{3}{6}=\tfrac12>0$.

Common Mistakes

Mistake 1: Using a negative scalar and calling it "same direction"

Why it's wrong: $k<0$ (e.g. $\vec b=-\vec a$) gives a vector along the same line but pointing exactly opposite — same collinearity, not the same direction. Correct approach: choose $k>0$ so the sense is preserved.

Mistake 2: Confusing "same direction" with "same magnitude"

Why it's wrong: same direction only fixes orientation; the two vectors should generally have different lengths (that's what makes them "different"). Correct approach: scale by a positive $k\neq1$ and check the unit vectors match.