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 non-zero vectors $a$ and $b$,if $|a+b| < |a-b|$,then $a$ and $b$ are

The components of a vector $a$ along and perpendicular to the non-zero vector $b$ are respectively:

The angle between the diagonals of the parallelogram whose adjacent sides are $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\hat{i}+2 \hat{j}+3 \hat{k}$ is

Let $v_1, v_2, v_3, v_4$ be unit vectors in the $XY$-plane,one each in the interior of the four quadrants. Which of the following statements is necessarily true?

If $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}$ and $|\overrightarrow{a}|=3, |\overrightarrow{b}|=4$ and $|\overrightarrow{c}|=\sqrt{37}$,then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is:

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?