Vector Algebra
Dot product using given vector conditions
nta_pyq_2023_jan
Grade 12
Question:
Let $\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}$, $\vec{b} = 3\hat{i} - 5\hat{j} - \lambda\hat{k}$, $\vec{a}\cdot\vec{c} = 7$, $2\vec{b}\cdot\vec{c} + 43 = 0$, $\vec{a}\times\vec{c} = \vec{b}\times\vec{c}$. Then $|\vec{a}\cdot\vec{b}|$ is equal to
Step-by-Step Solution
Key Concept: From $\vec{a}\times\vec{c} = \vec{b}\times\vec{c}$, deduce $(\vec{a}-\vec{b})\parallel\vec{c}$. Use $\vec{a}\cdot\vec{c}$ and $\vec{b}\cdot\vec{c}$ conditions to find $\lambda$.
$\mu=2, \lambda^2=1$. $\vec{a}\cdot\vec{b} = 3-10-\lambda^2 = 3-10-1=-8$. $|\vec{a}\cdot\vec{b}|=8$. Answer: 8
Correct Answer: 8