Animated Solution for Mathematics - Vector Algebra: Let $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. Let ${\vec{\beta }}_1$ be parallel to $\vec{\alpha }$ and ${\vec{\beta }}_2$ be perpendicular to $\vec{\alpha }$. If $\vec{\beta }={\vec{\beta }}_1+{\vec{\beta }}_2$, then the value of $5{\vec{\beta }}_2\cdot (\hat{i}+\hat{j}+\hat{k})$ is

Welcome, future engineer. Today, we are not just solving a problem; we are dissecting the anatomy of a vector. In the vast landscape of JEE Advanced physics and mathematics, the ability to resolve a vector into its parallel and perpendicular components is a superpower.

We are given a reference vector $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and a target vector $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. We aim to decompose $\vec{\beta }$ into two components: ${\vec{\beta }}_1$ (parallel to $\vec{\alpha }$) and ${\vec{\beta }}_2$ (perpendicular to $\vec{\alpha }$).

Since ${\vec{\beta }}_1$ is parallel to $\vec{\alpha }$, it must be a scalar multiple of the form ${\vec{\beta }}_1=\lambda \vec{\alpha }$. Given the vector addition $\vec{\beta }={\vec{\beta }}_1+{\vec{\beta }}_2$, we can define the perpendicular component as:

Substituting the given vectors into our expression for ${\vec{\beta }}_2$, we obtain:

Grouping the unit vectors, we get:

The constraint that ${\vec{\beta }}_2\perp \vec{\alpha }$ implies that their dot product must be zero, i.e., ${\vec{\beta }}_2\cdot \vec{\alpha }=0$. This yields the following equation:

Expanding the dot product by multiplying corresponding components, we have:

Distributing the constants, we get:

Combining the constant terms ($4+6-20=-10$) and the $\lambda$ terms ($-16\lambda -9\lambda -25\lambda =-50\lambda$), we arrive at:

With $\lambda =-\frac{1}{5}$, we reconstruct ${\vec{\beta }}_2$:

Simplifying the coefficients, we find:

The problem asks for the value of $5{\vec{\beta }}_2\cdot (\hat{i}+\hat{j}+\hat{k})$. Multiplying ${\vec{\beta }}_2$ by $5$ gives:

Finally, calculating the dot product with $(\hat{i}+\hat{j}+\hat{k})$:

The final result is 7.