Vector Algebra
Coplanar vectors; projection
nta_pyq_2023_jan
Grade 12
Question:
If the vectors $\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - 2\hat{k}$ and $\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda + \mu$ is equal to
Step-by-Step Solution
Key Concept: Coplanarity: scalar triple product $= 0$. Projection condition gives another equation. Solve for $\lambda, \mu$.
Coplanarity: $5\lambda-\mu=28$. Projection: $-2\lambda+4\mu-8 = \sqrt{54\times24} = 36 \Rightarrow -2\lambda+4\mu=44 \Rightarrow -\lambda+2\mu=22$. Solving: $\lambda+\mu=24$. Answer: (3)
Correct Answer: 24