Let vec{a}=hat{i}+2hat{j}+hat{k} and vec{b}=2 | vedclass

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

Question

(D) Let the vector $\vec{p}$ lie in the plane of $\vec{a}$ and $\vec{b}$. Then $\vec{p} = \vec{a} + \lambda \vec{b}$.Since $\vec{p}$ is perpendicular

Vedclass · Accepted Answer

$\frac{1}{\sqrt{2}}(-\hat{i}+\hat{k})$

Vedclass · Answer

(D) Let the vector $\vec{p}$ lie in the plane of $\vec{a}$ and $\vec{b}$. Then $\vec{p} = \vec{a} + \lambda \vec{b}$.Since $\vec{p}$ is perpendicular to $\vec{a}$,we have $\vec{p} \cdot \vec{a} = 0$.$(\vec{a} + \lambda \vec{b}) \cdot \vec{a} = 0 \Rightarrow \vec{a} \cdot \vec{a} + \lambda (\vec{b} \cdot \vec{a}) = 0$.Given $\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$.$\vec{a} \cdot \vec{a} = 1^2 + 2^2 + 1^2 = 6$.$\vec{b} \cdot \vec{a} = (2)(1) + (1)(2) + (-1)(1) = 2 + 2 - 1 = 3$.Substituting these values: $6 + \lambda(3) = 0 \Rightarrow \lambda = -2$.Thus,$\vec{p} = (\hat{i} + 2\hat{j} + \hat{k}) - 2(2\hat{i} + \hat{j} - \hat{k}) = -3\hat{i} + 3\hat{k}$.The unit vector $\hat{c}$ is $\pm \frac{\vec{p}}{|\vec{p}|} = \pm \frac{-3\hat{i} + 3\hat{k}}{\sqrt{(-3)^2 + 3^2}} = \pm \frac{-3\hat{i} + 3\hat{k}}{3\sqrt{2}} = \pm \frac{-\hat{i} + \hat{k}}{\sqrt{2}}$.Comparing with the options,the correct vector is $\frac{1}{\sqrt{2}}(-\hat{i} + \hat{k})$.

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is :

Similar Questions

The vector $\vec{a} = \alpha \hat{i} + 2\hat{j} + \beta \hat{k}$ lies in the plane of $\vec{b} = \hat{i} + \hat{j}$ and $\vec{c} = \hat{j} + \hat{k}$ and bisects the angle between $\vec{b}$ and $\vec{c}$. Find the possible values of $\alpha$ and $\beta$.

Let $a=\hat{i}+2 \hat{j}-2 \hat{k}$ and $b=2 \hat{i}-\hat{j}-2 \hat{k}$. If the orthogonal projection vector of $a$ on $b$ is $x$ and the orthogonal projection vector of $b$ on $a$ is $y$, then $|x-y|=$

If two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 4$, then $|\vec{a} - \vec{b}| = . . . . . . $.

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=3, |\vec{b}|=4, |\vec{c}|=5$ and each one of them is perpendicular to the sum of the other two. Find $|\vec{a}+\vec{b}+\vec{c}|$.

If $a, b$ and $c$ are mutually perpendicular vectors of the same magnitude,then the cosine of the angle between $a$ and $a+b+c$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module