Let bar{a}, bar{b}, bar{c} be three unit vect | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given that $\bar{a}, \bar{b}, \bar{c}$ are unit vectors, so $|\bar{a}| = |\bar{b}| = |\bar{c}| = 1$.The given expression is $|\bar{a}-\bar{b}|^2+|

Vedclass · Accepted Answer

Both Statement $I$ and Statement $II$ are false

Vedclass · Answer

(D) Given that $\bar{a}, \bar{b}, \bar{c}$ are unit vectors, so $|\bar{a}| = |\bar{b}| = |\bar{c}| = 1$.The given expression is $|\bar{a}-\bar{b}|^2+|\bar{a}-\bar{c}|^2=10$.Expanding this, we get:$(|\bar{a}|^2+|\bar{b}|^2-2\bar{a}\cdot\bar{b}) + (|\bar{a}|^2+|\bar{c}|^2-2\bar{a}\cdot\bar{c}) = 10$Since $|\bar{a}|=|\bar{b}|=|\bar{c}|=1$, we have:$(1+1-2\bar{a}\cdot\bar{b}) + (1+1-2\bar{a}\cdot\bar{c}) = 10$$4 - 2\bar{a}\cdot(\bar{b}+\bar{c}) = 10$$-2\bar{a}\cdot(\bar{b}+\bar{c}) = 6$$\bar{a}\cdot(\bar{b}+\bar{c}) = -3$ ... $(i)$For Statement $(I)$: $|\bar{a}+2 \bar{b}|^2+|2 \bar{a}+\bar{c}|^2$$= (|\bar{a}|^2 + 4|\bar{b}|^2 + 4\bar{a}\cdot\bar{b}) + (4|\bar{a}|^2 + |\bar{c}|^2 + 4\bar{a}\cdot\bar{c})$$= (1 + 4 + 4\bar{a}\cdot\bar{b}) + (4 + 1 + 4\bar{a}\cdot\bar{c})$$= 10 + 4(\bar{a}\cdot\bar{b} + \bar{a}\cdot\bar{c})$$= 10 + 4(\bar{a}\cdot(\bar{b}+\bar{c}))$$= 10 + 4(-3) = 10 - 12 = -2$.Since $-2 
eq 2$, Statement $(I)$ is false.For Statement $(II)$: $|2 \bar{a}+3 \bar{b}|^2+|3 \bar{a}+2 \bar{c}|^2$$= (4|\bar{a}|^2 + 9|\bar{b}|^2 + 12\bar{a}\cdot\bar{b}) + (9|\bar{a}|^2 + 4|\bar{c}|^2 + 12\bar{a}\cdot\bar{c})$$= (4 + 9 + 12\bar{a}\cdot\bar{b}) + (9 + 4 + 12\bar{a}\cdot\bar{c})$$= 26 + 12(\bar{a}\cdot(\bar{b}+\bar{c}))$$= 26 + 12(-3) = 26 - 36 = -10$.Since $-10 
eq 10$, Statement $(II)$ is false.Therefore, both statements are false.

Let $\bar{a}, \bar{b}, \bar{c}$ be three unit vectors satisfying $|\bar{a}-\bar{b}|^2+|\bar{a}-\bar{c}|^2=10$. Then
Statement $(I)$ : $|\bar{a}+2 \bar{b}|^2+|2 \bar{a}+\bar{c}|^2=2$.
Statement $(II)$ : $|2 \bar{a}+3 \bar{b}|^2+|3 \bar{a}+2 \bar{c}|^2=10$.
Which of the above statements is (are) true?

Similar Questions

For any two non-zero vectors $\vec{a}$ and $\vec{b}$,$(a \vec{b} + b \vec{a}) \cdot (a \vec{b} - b \vec{a})$ is equal to:

If $\overrightarrow{A} = \hat{i} + 2\hat{j} + 3\hat{k}$,$\overrightarrow{B} = -\hat{i} + 2\hat{j} + \hat{k}$ and $\overrightarrow{C} = 3\hat{i} + \hat{j}$,then the value of $t$ such that $\overrightarrow{A} + t\overrightarrow{B}$ is at a right angle to vector $3\hat{i} + 4\hat{j}$ is

If $\overrightarrow{x}$ and $\overrightarrow{y}$ are two non-zero vectors such that $|\overrightarrow{x}+\overrightarrow{y}|=|\overrightarrow{x}|$ and $2\overrightarrow{x}+\lambda\overrightarrow{y}$ is perpendicular to $\overrightarrow{y}$,then the value of $\lambda$ is

The vectors $3 \vec{a}-5 \vec{b}$ and $2 \vec{a}+\vec{b}$ are mutually perpendicular and the vectors $\vec{a}+4 \vec{b}$ and $-\vec{a}+\vec{b}$ are also mutually perpendicular. Then the angle between the vectors $\vec{a}$ and $\vec{b}$ is:

If $ABCD$ is a cyclic quadrilateral with $R$ as the radius of the circumcircle and $(AB)^2+(CD)^2=4R^2$,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

Let $\bar{a}, \bar{b}, \bar{c}$ be three unit vectors satisfying $|\bar{a}-\bar{b}|^2+|\bar{a}-\bar{c}|^2=10$. ThenStatement $(I)$ : $|\bar{a}+2 \bar{b}|^2+|2 \bar{a}+\bar{c}|^2=2$.Statement $(II)$ : $|2 \bar{a}+3 \bar{b}|^2+|3 \bar{a}+2 \bar{c}|^2=10$.Which of the above statements is (are) true?