Vectors
Non-Collinear Vectors
MMTS_Full_Test_16
Grade 12
Question:
Let $\vec{a},\vec{b}$ be two non-collinear unit vectors and $\vec{x}=\vec{a}-(\vec{a}\cdot\vec{b})\vec{b}$ and $\vec{y}=\vec{a}\times\vec{b}$. Statement-I: $|\vec{x}|=|\vec{y}|$. Statement-II: $|\vec{y}|=|\vec{x}|+|\vec{x}\cdot\vec{b}|$.
S-I is true but S-II is false
Both S-I and S-II are false
Both S-I and S-II are true
S-I is false but S-II is true
Step-by-Step Solution
Key Concept: $\vec{x}$ is projection of $\vec{a}$ perpendicular to $\vec{b}$; $\vec{y}=\vec{a}\times\vec{b}$
$|\vec{x}|^2=1-(\vec{a}\cdot\vec{b})^2=\sin^2\theta=|\vec{y}|^2$. S-I true. S-II: $|\vec{y}|=|\vec{x}|$ and $|\vec{x}\cdot\vec{b}|=0$, so $|\vec{y}|=|\vec{x}|+0=|\vec{x}|$: true. Both true.
Correct Answer: 3