Let a and b be linearly independent vectors s | vedclass

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

Question

(D) Given $|a| = \sqrt{3}$, $|b| = 3$, and $|a - b| = 4$. Squaring $|a - b| = 4$ gives $|a|^2 + |b|^2 - 2(a \cdot b) = 16$, so $3 + 9 - 2(a \cdot b) =

Vedclass · Accepted Answer

$128$

Vedclass · Answer

(D) Given $|a| = \sqrt{3}$, $|b| = 3$, and $|a - b| = 4$. Squaring $|a - b| = 4$ gives $|a|^2 + |b|^2 - 2(a \cdot b) = 16$, so $3 + 9 - 2(a \cdot b) = 16$, which implies $a \cdot b = -2$.Let $v = 2i + 2j - k$. The equation $a 	imes v = v 	imes b$ can be written as $a 	imes v + b 	imes v = 0$, so $(a + b) 	imes v = 0$.This implies $(a + b)$ is parallel to $v = 2i + 2j - k$. Thus, $a + b = k(2i + 2j - k)$ for some scalar $k$.Calculate $|a + b|^2 = |a|^2 + |b|^2 + 2(a \cdot b) = 3 + 9 + 2(-2) = 8$.Since $|a + b|^2 = k^2(2^2 + 2^2 + (-1)^2) = 9k^2$, we have $9k^2 = 8$, so $k^2 = 8/9$ and $|k| = \frac{2\sqrt{2}}{3}$.We need $|(a + b) \cdot (3i + 4j + 2k)| = |k(2i + 2j - k) \cdot (3i + 4j + 2k)| = |k(6 + 8 - 2)| = |12k| = 12 \cdot \frac{2\sqrt{2}}{3} = 8\sqrt{2}$.Given this equals $\sqrt{\lambda}$, so $\sqrt{\lambda} = 8\sqrt{2} = \sqrt{64 \cdot 2} = \sqrt{128}$. Thus $\lambda = 128$.

Let $a$ and $b$ be linearly independent vectors such that $|a| = \sqrt{3}$, $|b| = 3$, and $|a - b| = 4$. If $a \times (2i + 2j - k) = (2i + 2j - k) \times b$ and $|(a + b) \cdot (3i + 4j + 2k)| = \sqrt{\lambda}$, then $\lambda = ...$

Similar Questions

$p=2 \hat{i}-3 \hat{j}+\hat{k}, q=\hat{i}+\hat{j}-\hat{k}$. If the vectors $a$ and $b$ are the orthogonal projections of $p$ on $q$ and $q$ on $p$ respectively, then $\frac{a \times b}{a \cdot b}=$

Let $u, v$ and $w$ be three vectors such that $u+v+w=0$,$|u|=3$,$|v|=5$ and $|w|=7$. Then the angle between $u$ and $v$ is (in $^{\circ}$)

If $\bar{a}=\hat{i}+2 \hat{j}-3 \hat{k}$,$\bar{b}=3 \hat{i}-\hat{j}+2 \hat{k}$,$\bar{c}=\hat{i}+3 \hat{j}+\hat{k}$ and $\bar{a}+\lambda \bar{b}$ is perpendicular to $\bar{c}$,then $\lambda=$

The unit vector in $ZOX$ plane, making angles $45^{\circ}$ and $60^{\circ}$ respectively with $\vec{\alpha}=2 \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{\beta}=\hat{j}-\hat{k}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module