a=3 hat{i}+hat{j}-hat{k}, b=hat{i}-4 hat{j}+5 | vedclass

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

Question

(A) Given vectors $a=3 \hat{i}+\hat{j}-\hat{k}, b=\hat{i}-4 \hat{j}+5 \hat{k}$ and $c=4 \hat{i}+5 \hat{j}-\hat{k}$.Since $r$ is perpendicular to both

Vedclass · Accepted Answer

$3(\hat{i}-\hat{j}-\hat{k})$

Vedclass · Answer

(A) Given vectors $a=3 \hat{i}+\hat{j}-\hat{k}, b=\hat{i}-4 \hat{j}+5 \hat{k}$ and $c=4 \hat{i}+5 \hat{j}-\hat{k}$.Since $r$ is perpendicular to both $b$ and $c$,$r$ must be parallel to $b 	imes c$.$b 	imes c = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 1 & -4 & 5 \ 4 & 5 & -1 \end{vmatrix} = \hat{i}(4-25) - \hat{j}(-1-20) + \hat{k}(5+16) = -21 \hat{i} + 21 \hat{j} + 21 \hat{k} = 21(-\hat{i} + \hat{j} + \hat{k})$.Let $r = \lambda(-\hat{i} + \hat{j} + \hat{k})$ for some scalar $\lambda$.Given $r \cdot a = 9$,we have $\lambda(-\hat{i} + \hat{j} + \hat{k}) \cdot (3 \hat{i} + \hat{j} - \hat{k}) = 9$.$\lambda(-3 + 1 - 1) = 9 \Rightarrow -3\lambda = 9 \Rightarrow \lambda = -3$.Thus,$r = -3(-\hat{i} + \hat{j} + \hat{k}) = 3(\hat{i} - \hat{j} - \hat{k})$.Therefore,option $A$ is correct.

$a=3 \hat{i}+\hat{j}-\hat{k}, b=\hat{i}-4 \hat{j}+5 \hat{k}, c=4 \hat{i}+5 \hat{j}-\hat{k}$ are three vectors and a vector $r$ is perpendicular to both the vectors $b$ and $c$. If $r \cdot a=9$,then $r=$

Similar Questions

Let $\vec{a} = \alpha \hat{i} + \hat{j} - \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{j} - \alpha \hat{k}$,where $\alpha > 0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $\vec{c} = -\hat{i} + 2 \hat{j} - 2 \hat{k}$ is $30$,then $\alpha$ is equal to:

Let $\hat{u} = u_1 \hat{i} + u_2 \hat{j} + u_3 \hat{k}$ be a unit vector in $\mathbb{R}^3$ and $\hat{v} = \frac{1}{\sqrt{6}}(\hat{i} + \hat{j} + 2 \hat{k})$. Given that there exists a unit vector $\vec{w}$ such that $\hat{u} \times \vec{w} = \hat{v}$,which of the following is(are) correct?

Let $\bar{a}$,$\bar{b}$,and $\bar{c}$ be unit vectors. Suppose that $\bar{a} \cdot \bar{b} = \bar{a} \cdot \bar{c} = 0$ and the angle between $\bar{b}$ and $\bar{c}$ is $\frac{\pi}{6}$. Then $\bar{a}$ is equal to:

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them,then for what value of $\theta$ is $2\hat{u} \times 3\hat{v}$ a unit vector?

Let $p, q$ and $r$ be vectors such that $r \neq 0$,$p \times q = r$,and $q \times p = r$. Then which of the following is true?
$(i)$ $p, q, r$ are pair-wise orthogonal vectors
(ii) $|q| = |r| = |p|$

Vedclass Test Series

Exam Paper Generator

Online Exam Module