If the vectors vec{a} = hat{i} - hat{j} + 2ha | vedclass

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

Question

(D) Given vectors are $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$,$\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}$,and $\vec{c} = \alpha\hat{i} + \hat{j} + \bet

Vedclass · Accepted Answer

$(-3, 2)$

Vedclass · Answer

(D) Given vectors are $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$,$\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}$,and $\vec{c} = \alpha\hat{i} + \hat{j} + \beta\hat{k}$.Since the vectors are mutually orthogonal,their dot products are zero: $\vec{a} \cdot \vec{c} = 0$ and $\vec{b} \cdot \vec{c} = 0$.For $\vec{a} \cdot \vec{c} = 0$: $(\hat{i} - \hat{j} + 2\hat{k}) \cdot (\alpha\hat{i} + \hat{j} + \beta\hat{k}) = \alpha - 1 + 2\beta = 0 \implies \alpha + 2\beta = 1$ (Equation $1$).For $\vec{b} \cdot \vec{c} = 0$: $(2\hat{i} + 4\hat{j} + \hat{k}) \cdot (\alpha\hat{i} + \hat{j} + \beta\hat{k}) = 2\alpha + 4 + \beta = 0 \implies 2\alpha + \beta = -4$ (Equation $2$).From Equation $2$,$\beta = -4 - 2\alpha$. Substituting this into Equation $1$:$\alpha + 2(-4 - 2\alpha) = 1 \implies \alpha - 8 - 4\alpha = 1 \implies -3\alpha = 9 \implies \alpha = -3$.Substituting $\alpha = -3$ into $\beta = -4 - 2\alpha$: $\beta = -4 - 2(-3) = -4 + 6 = 2$.Thus,$(\alpha, \beta) = (-3, 2)$.

If the vectors $\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$,$\vec{b} = 2\hat{i} + 4\hat{j} + \hat{k}$,and $\vec{c} = \alpha\hat{i} + \hat{j} + \beta\hat{k}$ are mutually orthogonal,then $(\alpha, \beta) = $

Similar Questions

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

Find a unit vector in the $xy$-plane which makes an angle of $45^{\circ}$ with the vector $i + j$ and an angle of $60^{\circ}$ with the vector $3i - 4j$.

The magnitude of the projection of vector $\vec{a} = -\hat{i} + 2\hat{j} - \hat{k}$ on the vector $\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}$ is . . . . . . .

Let $ABC$ be a triangle and $P$ be a point inside $ABC$ such that $\overrightarrow{PA} + 2\overrightarrow{PB} + 3\overrightarrow{PC} = \vec{0}$. The ratio of the area of $\triangle ABC$ to that of $\triangle APC$ is

Let $a, b$ and $c$ be unit vectors such that $a$ is perpendicular to the plane containing $b$ and $c$ and the angle between $b$ and $c$ is $\frac{\pi}{3}$. Then,$|a+b+c|=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module