Vector Algebra
Projection decomposition of a vector
nta_pyq_2023_jan
Grade 12
Question:
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
Step-by-Step Solution
Key Concept: Decompose $\vec{\beta}$ into components parallel and perpendicular to $\vec{\alpha}$ using projection formula.
$\lambda = \frac{\vec{\alpha}\cdot\vec{\beta}}{|\vec{\alpha}|^2} = \frac{4+6-20}{50} = -\frac{10}{50} = -\frac{1}{5}$. $5\vec{\beta}_2 = 5\vec{\beta} - 5\lambda\vec{\alpha} = (5+4)\hat{i}+(10+3)\hat{j}+(-20+5)\hat{k} = 9\hat{i}+13\hat{j}-15\hat{k}$. $5\vec{\beta}_2\cdot(\hat{i}+\hat{j}+\hat{k}) = 9+13-15=7$. Answer: (3)
Correct Answer: 7