Vector Algebra
Projection of a vector on another; system of equations
nta_pyq_2023_jan
Grade 12
Question:
Let $\vec{a} = 4\hat{i}+3\hat{j}$ and $\vec{b} = 3\hat{i}-4\hat{j}+5\hat{k}$ and $\vec{c}$ is a vector such that $\vec{c}\cdot(\vec{a}\times\vec{b})+25=0$, $\vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})=4$ and projection of $\vec{c}$ on $\vec{a}$ is 1, then the projection of $\vec{c}$ on $\vec{b}$ equals:
$\frac{5}{\sqrt{2}}$
$\frac{1}{5}$
$\frac{1}{\sqrt{2}}$
$\frac{3}{\sqrt{2}}$
Step-by-Step Solution
Key Concept: Set up three equations for $\vec{c}=(x,y,z)$ using the given conditions, solve, then compute projection.
Solving: $\vec{c} = 2\hat{i}-\hat{j}+3\hat{k}$. Projection on $\vec{b} = \frac{\vec{c}\cdot\vec{b}}{|\vec{b}|} = \frac{6+4+15}{5\sqrt{2}} = \frac{25}{5\sqrt{2}} = \frac{5}{\sqrt{2}}$. Answer: (1)
Correct Answer: $\frac{5}{\sqrt{2}}$