Animated Solution for Mathematics - Vector Algebra: If vectors ${\vec{a}}_1=x\hat{i}-\hat{j}+\hat{k}$ and ${\vec{a}}_2=\hat{i}+y\hat{j}+z\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i}+y\hat{j}+z\hat{k}$ is :

Welcome, fellow traveler on the JEE journey! Today, we are going to unravel a beautiful problem involving vectors. It might look like a simple algebraic exercise, but beneath the surface lies a fundamental geometric truth.

We are given two vectors, ${\vec{a}}_1=x\hat{i}-\hat{j}+\hat{k}$ and ${\vec{a}}_2=\hat{i}+y\hat{j}+z\hat{k}$, and we are told they are collinear.

If two vectors are collinear, they lie on the same line and are essentially parallel. Mathematically, this means one vector is just a scaled version of the other: This relationship is our key to the kingdom.

When we write ${\vec{a}}_1=\lambda {\vec{a}}_2$, we are stating that the components of ${\vec{a}}_1$ are the components of ${\vec{a}}_2$ multiplied by $\lambda$. This yields the following system of equations:

We can rewrite these as ratios to isolate the parameter $\lambda$:

By equating each term to $\lambda$, we express our unknowns in terms of this single parameter: This is a classic JEE strategy: whenever you see multiple variables, look for a way to parameterize them to reduce the complexity of the problem.

Now, let's examine the target vector $\vec{v}=x\hat{i}+y\hat{j}+z\hat{k}$. Substituting our expressions for $x$, $y$, and $z$, we get:

Notice something fascinating about the components. Since $y=-\frac{1}{\lambda }$ and $z=\frac{1}{\lambda }$, it follows that $y=-z$.

Furthermore, consider the product of $x$ and $z$: Since their product is positive, $x$ and $z$ must share the same sign. We have uncovered the DNA of our target vector: $y$ must be the negative of $z$, and $x$ and $z$ must have the same sign.

We are almost there! We need to check our options against these two rules: $y=-z$ and $x,z$ having the same sign.

Let's test the vector $\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$. Here, the components are proportional to $1,-1,1$.

Does it satisfy our rules? Yes: 1. $y=-1$ and $z=1$, so $y=-z$ holds. 2. $x=1$ and $z=1$, so they have the same sign.

If we set $\lambda =1$, our vector $\vec{v}$ becomes $\hat{i}-\hat{j}+\hat{k}$. The magnitude is:

Dividing by the magnitude gives us exactly $\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$. We have solved it! Remember, in JEE, the math is just a language; once you translate the problem, the solution reveals itself.